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A proper $k$-coloring of $G$ is called an odd coloring of $G$ if for every vertex $v$, there is a color that appears at an odd number of neighbors of $v$. This concept was introduced recently by Petru\v{s}evski and \v{S}krekovski, and they conjectured that every planar graph is odd 5-colorable. Towards this conjecture, Caro, Petru\v{s}evski, and \v{S}krekovski showed that every outerplanar graph is odd 5-colorable, and this bound is tight since the cycle of length 5 is not odd 4-colorable. Recently, the first author and others showed that every maximal outerplanar graph is odd 4-colorable. In this paper, we show that a connected outerplanar graph $G$ is odd 4-colorable if and only if $G$ contains a block which is not a copy of the cycle of length 5. This strengthens the result by Caro, Petru\v{s}evski, and \v{S}krekovski, and gives a complete characterization of odd 4-colorable outerplanar graphs., Comment: 9 pages

Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to N$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is $k$-truncated-degree-choosable. It was proved in [Zhou,Zhu,Zhu, Arc-weighted acyclic orientations and variations of degeneracy of graphs, arXiv:2308.15853] that there is a 3-connected non-complete planar graph that is not 7-truncated-degree-choosable, and every 3-connected non-complete planar graph is 16-truncated-degree-choosable. This paper improves the bounds, and proves that there is a 3-connected non-complete planar graph that is not 8-truncated-degree-choosable and every non-complete 3-connected planar graph is 12-truncated-degree-choosable., Comment: 17 pages, 1 figure

Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert. We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \in \mathbb{R}_{\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\mathbb{R}_{\geq 0}^{V(G)}$., Comment: 19 pages, 2 figures

One of Thomassen's classical results is that every planar graph of girth at least $5$ is 3-choosable. One can wonder if for a planar graph $G$ of girth sufficiently large and a $3$-list-assignment $L$, one can do even better. Can one find $3$ disjoint $L$-colorings (a packing), or $2$ disjoint $L$-colorings, or a collection of $L$-colorings that to every vertex assigns every color on average in one third of the cases (a fractional packing)? We prove that the packing is impossible, but two disjoint $L$-colorings are guaranteed if the girth is at least $8$, and a fractional packing exists when the girth is at least $6.$ For a graph $G$, the least $k$ such that there are always $k$ disjoint proper list-colorings whenever we have lists all of size $k$ associated to the vertices is called the list packing number of $G$. We lower the two-times-degeneracy upper bound for the list packing number of planar graphs of girth $3,4$ or $5$. As immediate corollaries, we improve bounds for $\epsilon$-flexibility of classes of planar graphs with a given girth. For instance, where previously Dvo\v{r}\'{a}k et al. proved that planar graphs of girth $6$ are (weighted) $\epsilon$-flexibly $3$-choosable for an extremely small value of $\epsilon$, we obtain the optimal value $\epsilon=\frac{1}{3}$. Finally, we completely determine and show interesting behavior on the packing numbers for $H$-minor-free graphs for some small graphs $H.$, Comment: 36 pages, 8 figures

Assume G is a graph and k is a positive integer. Let f: V(G) to N be defined as f(v)=min{k, d_G(v)}. If G is DP-f-colourable (respectively, f-choosable), then we say G is DP-k-truncated-degree-colourable (respectively, k-truncated-degree-choosable). Hutchinson [On list-colouring outerplanar graphs. J. Graph Theory] proved that 2-connected maximal outerplanar graphs other than the triangle are 5-truncated-degree-choosable. This result was recently improved by Dai, Hu, Li, and Maezawa in [On DP-colouring of outerplanar graphs. Manuscript, 2023], where it is proved that 2-connected outerplanar graphs other than cycles are DP-5-truncated-degree-colourable. This paper further improves this result and proves that 2-connected K_{2,4}-minor free graphs other than cycles and complete graphs are DP-$5$-truncated-degree-colourable., Comment: 25 pages, 9 figures

A graph $G$ is $(1,3)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ so that every vertex in $G[V_1]$ has degree at most $1$ and every vertex in $G[V_2]$ has degree at most $3$. We prove that every graph with maximum average degree at most 28/9 is $(1, 3)$-colorable., Comment: 21 pages

This paper studies generalizations of the concept of acyclic orientations to arc-weighted orientations. These lead to four types of variations of strict degeneracy of graphs. Some of these variations are studied in the literature under different names and we put them in a same framework for comparison. Then we concentrate on one of these variations, which is new and is defined as follows: For a graph $G$ and a mapping $f \in \mathbb{N}^G$, we say $G$ is $ST^{(2)}$-$f$-degenerate if there is an arc-weighted orientation $(D, w)$ of $G$ such that $d_{(D,w)}^+(v) < f(v)$ for each vertex $v$, and every sub-digraph $D'$ of $D$ contains an arc $e=(u,v)$ with $w(e) > d_{(D', w)}^+(v)$. We prove that if $G$ is $ST^{(2)}$-$f$-degenerate, then $G$ is $f$-paintable, as well as $f$-AT. Then we use $ST^{(2)}$-degeneracy to study truncated degree choosability of graphs. A graph $G$ is called $k$-truncated degree-choosable (respectively $ST^{(2)}$-$k$-truncated degree degenerate) if $G$ is $f$-choosable (respectively, $ST^{(2)}$-$f$-degenerate), where $f(v)= \min\{k, d_G(v)\}$. Richter asked whether every 3-connected non-complete planar graph is $6$-truncated-degree-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not $7$-truncated-degree-choosable. On the other hand, we prove that every 3-connected non-complete planar graph is $ST^{(2)}$-$16$-truncated-degree-degenerate, and hence $16$-truncated-degree-choosable. We further prove that for an arbitrary proper minor closed family ${\mathcal G}$ of graphs, let $s$ be the minimum integer such that $K_{s,t} \notin \mathcal{G}$ for some $t$, then there is a constant $k$ such that every $s$-connected non-complete graph $G \in {\mathcal G}$ is $ST^{(2)}$-$k$-truncated-degree-degenerate and hence $k$-truncated-degree-choosable., Comment: 24 pages, 1 figure

Given positive integers $p \ge k$, and a non-negative integer $d$, we say a graph $G$ is $(k,d,p)$-choosable if for every list assignment $L$ with $|L(v)|\geq k$ for each $v \in V(G)$ and $|\bigcup_{v\in V(G)}L(v)| \leq p$, there exists an $L$-coloring of $G$ such that each monochromatic subgraph has maximum degree at most $d$. In particular, $(k,0,k)$-choosable means $k$-colorable, $(k,0,+\infty)$-choosable means $k$-choosable and $(k,d,+\infty)$-choosable means $d$-defective $k$-choosable. This paper proves that there are 1-defective 3-choosable graphs that are not 4-choosable, and for any positive integers $\ell \geq k \geq 3$, and non-negative integer $d$, there are $(k,d, \ell)$-choosable graphs that are not $(k,d , \ell+1)$-choosable. These results answer questions asked by Wang and Xu [SIAM J. Discrete Math. 27, 4(2013), 2020-2037], and Kang [J. Graph Theory 73, 3(2013), 342-353], respectively. Our construction of $(k,d, \ell)$-choosable but not $(k,d , \ell+1)$-choosable graphs generalizes the construction of Kr\'{a}l' and Sgall in [J. Graph Theory 49, 3(2005), 177-186] for the case $d=0$., Comment: 12 pages, 4 figures

Weak degeneracy is a variation of degeneracy which shares many nice properties of degeneracy. In particular, if a graph $G$ is weakly $d$-degenerate, then for any $(d + 1)$-list assignment $L$ of $G$, one can construct an $L$-coloring of $G$ by a modified greedy coloring algorithm. It is known that planar graphs of girth 5 are 3-choosable and locally planar graphs are 5-choosable. This paper strengthens these results and proves that planar graphs of girth 5 are weakly 2-degenerate and locally planar graphs are weakly 4-degenerate., Comment: 13pages

This paper proves that every planar graph without cycles of length 4, 7, or 9 is DP-3-colorable., Comment: 12 pages, 2 figures

This paper studies two variants of defective acyclic coloring of planar graphs. For a graph $G$ and a coloring $\varphi$ of $G$, a 2CC transversal is a subset $E'$ of $E(G)$ that intersects every 2-colored cycle. Let $k$ be a positive integer. We denote by $m_k(G)$ the minimum integer $m$ such that $G$ has a proper $k$-coloring which has a 2CC transerval of size $m$, and by $m'_k(G)$ the minimum size of a subset $E'$ of $E(G)$ such that $G-E'$ is acyclic $k$-colorable. We prove that for any $n$-vertex $3$-colorable planar graph $G$, $m_3(G) \le n - 3$ and for any planar graph $G$, $m_4(G) \le n - 5$ provided that $n \ge 5$. We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal $E'$ can be chosen in such a way that $E'$ induces a forest. We also prove that for any planar graph $G$, $m'_3(G) \le (13n - 42) / 10$ and $m'_4(G) \le (3n - 12) / 5$.

In this paper the concept of circular $r$-flows in a mono-directed signed graph $(G, \sigma)$ is introduced. That is a pair $(D, f)$, where $D$ is an orientation on $G$ and $f: E(G)\to (-r,r)$ satisfies that $|f(e)|\in [1, r-1]$ for each positive edge $e$ and $|f(e)|\in [0, \frac{r}{2}-1]\cup [\frac{r}{2}+1, r)$ for each negative edge $e$, and the total in-flow equals the total out-flow at each vertex. The circular flow index of a signed graph $(G, \sigma)$ with no positive bridge, denoted $\Phi_c(G,\sigma)$, is the minimum $r$ such that $(G, \sigma)$ admits a circular $r$-flow. This is the dual notion of circular colorings and circular chromatic numbers of signed graphs recently introduced in [Circular chromatic number of signed graphs. R. Naserasr, Z. Wang, and X. Zhu. Electronic Journal of Combinatorics, 28(2)(2021), \#P2.44], and is distinct from the concept of circular flows in bi-directed graphs associated to signed graphs studied in the literature. We give several equivalent definitions, study basic properties of circular flows in mono-directed signed graphs, explore relations with flows in graphs, and focus on upper bounds on $\Phi_c(G,\sigma)$ in terms of the edge-connectivity of $G$. Meanwhile, we note that for the particular values of $r_{_k}=\frac{2k}{k-1}$, and when restricted to two natural subclasses of signed graphs, the existence of a circular $r_{_k} $-flow is strongly connected with the existence of a modulo $k$-orientation, and in case of planar graphs, based on duality, with the homomorphisms to $C_{-k}$.

A graph is $\ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $\ell$ vertices. We prove that strongly regular graphs with at least six vertices are $2$-reconstructible., Comment: 5 pages

Assume $\lambda=\{k_1,k_2, \ldots, k_q\}$ is a partition of $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a $k_\lambda$-list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into $|\lambda|= q$ sets $C_1,C_2,\ldots,C_q$ such that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is \emph{$\lambda$-choosable} if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. The concept of $\lambda$-choosability is a refinement of choosability that puts $k$-choosability and $k$-colourability in the same framework. If $|\lambda|$ is close to $k_\lambda$, then $\lambda$-choosability is close to $k_\lambda$-colourability; if $|\lambda|$ is close to $1$, then $\lambda$-choosability is close to $k_\lambda$-choosability. This paper studies Hadwiger's Conjecture in the context of $\lambda$-choosability. Hadwiger's Conjecture is equivalent to saying that every $K_t$-minor-free graph is $\{1 \star (t-1)\}$-choosable for any positive integer $t$. We prove that for $t \ge 5$, for any partition $\lambda$ of $t-1$ other than $\{1 \star (t-1)\}$, there is a $K_t$-minor-free graph $G$ that is not $\lambda$-choosable. We then construct several types of $K_t$-minor-free graphs that are not $\lambda$-choosable, where $k_\lambda - (t-1)$ gets larger as $k_\lambda-|\lambda|$ gets larger. In partcular, for any $q$ and any $\epsilon > 0$, there exists $t_0$ such that for any $t \ge t_0$, for any partition $\lambda$ of $\lfloor (2-\epsilon)t \rfloor$ with $|\lambda| =q$, there is a $K_t$-minor-free graph that is not $\lambda$-choosable. The $q=1$ case of this result was recently proved by Steiner, and our proof uses a similar argument. We also generalize this result to $(a,b)$-list colouring.

For a multi-set $\lambda=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. The concept of $\lambda$-choosability puts $k$-colourability and $k$-choosability in the same framework: If $\lambda = \{k\}$, then $\lambda$-choosability is equivalent to $k$-choosability; if $\lambda$ consists of $k $ copies of $1$, then $\lambda$-choosability is equivalent to $k $-colourability. If $G$ is $\lambda$-choosable, then $G$ is $k_{\lambda}$-colourable. On the other hand, there are $k_{\lambda}$-colourable graphs that are not $\lambda$-choosable, provided that $\lambda$ contains an integer larger than $1$. Let $\phi(\lambda)$ be the minimum number of vertices in a $k_{\lambda}$-colourable non-$\lambda$-choosable graph. This paper determines the value of $\phi(\lambda)$ for all $\lambda$., Comment: 13 pages

A decomposition of a graph $G$ is a family of subgraphs of $G$ whose edge sets form a partition of $E(G)$. In this paper, we prove that every triangle-free planar graph $G$ can be decomposed into a $2$-degenerate graph and a matching. Consequently, every triangle-free planar graph $G$ has a matching $M$ such that $G-M$ is online 3-DP-colorable. This strengthens an earlier result in [R. \v{S}krekovski, {\em A Gr\"{o}tzsch-Type Theorem for List Colourings with Impropriety One}, Combin. Prob. Comput. 8 (1999), 493-507] that every triangle-free planar graph is $1$-defective $3$-choosable.

This paper proves the following result: If $G$ is a planar graph and $L$ is a $4$-list assignment of $G$ such that $|L(x) \cap L(y)| \le 2$ for every edge $xy$, then $G$ is $L$-colourable. This answers a question asked by Kratochv\'{i}l, Tuza and Voigt in [Journal of Graph Theory, 27(1):43--49, 1998]., Comment: 11 pages

This paper gives a simple proof of the result that every planar graph $G$ has Alon-Tarsi number at most 5, and has a matching $M$ such that $G-M$ has Alon-Tarsi number at most 4., Comment: 4 pages

It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are chromatic-choosable. This upper bound on $|V(G)|$ is tight: if $k$ is even, then $K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $2 k+2$ vertices that are not chromatic-choosable. It was proved in [arXiv:2201.02060] that these are the only non-$k$-choosable complete $k$-partite graphs with $2k+2$ vertices. For $G =K_{3 \star (k/2+1), 1 \star (k/2-1)}$ or $K_{4, 2 \star (k-1)}$, a bad list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that $G$ is not $L$-colourable. Bad list assignments for $G=K_{4, 2 \star (k-1)}$ were characterized in [Discrete Mathematics 244 (2002), 55-66]. In this paper, we first give a simpler proof of this result, and then we characterize bad list assignments for $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$. Using these results, we characterize all non-$k$-choosable (non-complete) $k$-partite graphs with $2k+2$ vertices., Comment: 13 pages, 1 figure

The concept of DP-coloring of a graph is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. Multiple DP-coloring of graphs, as a generalization of multiple list coloring, was first studied by Bernshteyn, Kostochka and Zhu in 2019. This paper proves that planar graphs without 3-cycles and normally adjacent 4-cycles are $(7m, 2m)$-DP-colorable for every integer $m$. As a consequence, the strong fractional choice number of any planar graph without 3-cycles and normally adjacent 4-cycles is at most $7/2$., Comment: 22 pages

A graph $G$ is called chromatic-choosable if $\chi(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are $k$-choosable. This upper bound on $|V(G)|$ is tight. It is known that if $k$ is even, then $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $G=K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $|V(G)| =2 k+2$ that are not $k$-choosable. Some subgraphs of these two graphs are also non-$k$-choosable. The main result of this paper is that all other $k$-chromatic graphs $G$ with $|V(G)| =2 k+2$ are $k$-choosable. In particular, if $\chi(G)$ is odd and $|V(G)| \le 2\chi(G)+2$, then $G$ is chromatic-choosable, which was conjectured by Noel., Comment: 33 pages

A $(d,h)$-decomposition of a graph $G$ is an ordered pair $(D, H)$ such that $H$ is a subgraph of $G$ of maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ with maximum out-degree at most $d$. In this paper, we prove that for $l \in \{5, 6, 7, 8, 9\}$, every planar graph without $4$- and $l$-cycles is $(2,1)$-decomposable. As a consequence, for every planar graph $G$ without $4$- and $l$-cycles, there exists a matching $M$, such that $G - M$ is $3$-DP-colorable and has Alon-Tarsi number at most $3$. In particular, $G$ is $1$-defective $3$-DP-colorable, $1$-defective $3$-paintable and 1-defective 3-choosable. These strengthen the results in [Discrete Appl. Math. 157~(2) (2009) 433--436] and [Discrete Math. 343 (2020) 111797]., Comment: 16 pages

This paper studies the strong fractional choice number $ch^s_f(G)$ and the strong fractional paint number $\chi^s_{f,P}(G)$ of a graph $G$. We prove that these parameters of any finite graph are rational numbers. On the other hand, for any positive integers $p,q$ satisfying $2 \le \frac{2p}{2q+1} \leq \lfloor\frac{p}{q}\rfloor$, there exists a graph $G$ with $ch^s_f(G) = \chi^s_{f,P}(G) = \frac{p}{q}$. The relationship between $\chi^s_{f,P}(G)$ and $ch^s_f(G)$ is explored. We prove that the gap $\chi^s_{f,P}(G)-ch^s_f(G)$ can be arbitrarily large. The strong fractional choice number of a family $\mathcal{G}$ of graphs is the supremum of the strong fractional choice number of graphs in $\mathcal{G}$. Let $\mathcal{P}$ denote the class of planar graphs and $\mathcal{P}_{k_1,\ldots, k_q}$ denote the class of planar graphs without $k_i$-cycles for $i=1,\ldots, q$. We prove that $3 + \frac{1}{2} \leq ch^s_f(\mathcal{P}_{ 4}) \leq 4$, $ch^s_f(\mathcal{P}_{ k})=4$ for $k \in \{5,6\}$, $3 +\frac{1}{12} \leq ch^s_f(\mathcal{P}_{ 4,5}) \leq 4$ and $ch^s_f(\mathcal{P}) \ge 4+\frac 13$. The last result improves the lower bound $4+\frac 29$ in [X. Zhu, multiple list colouring of planar graphs, Journal of Combin. Th. Ser. B,122(2017),794-799]., Comment: 20 pages,6 figures

A graph $G$ is total weight $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a proper total $L$-weighting, i.e., a mapping $f: V(G) \cup E(G) \to \mathbb{R}$ such that for each $z \in V(G) \cup E(G)$, $f(z) \in L(z)$, and for each edge $uv$ of $G$, $\sum_{e \in E(u)}f(e)+f(u) \ne \sum_{e \in E(v)}f(e) + f(v)$. This paper proves that if $G$ decomposes into complete graphs of odd order, then $G$ is total weight $(1,3)$-choosable. As a consequence, every Eulerian graph $G$ of large order and with minimum degree at least $0.91|V(G)|$ is total weight $(1,3)$-choosable. We also prove that any graph $G$ with minimum degree at least $0.999|V(G)|$ is total weight $(1,4)$-choosable., Comment: 10 pages. arXiv admin note: text overlap with arXiv:2104.05410

Assume $ k $ is a positive integer, $ \lambda=\{k_1,k_2,...,k_q\} $ is a partition of $ k $ and $ G $ is a graph. A $\lambda$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ \bigcup_{v\in V(G)} L(v) $ can be partitioned into $ q $ subsets $ C_1\cup C_2\cup\cdots\cup C_q $ and for each vertex $ v $ of $ G $, $ |L(v)\cap C_i|=k_i $. We say $ G $ is $\lambda$-choosable if for each $\lambda$-assignment $ L $ of $ G $, $ G $ is $ L $-colourable. In particular, if $ \lambda=\{k\} $, then $\lambda$-choosable is the same as $ k $-choosable, if $ \lambda=\{1, 1,...,1\} $, then $\lambda$-choosable is equivalent to $ k $-colourable. For the other partitions of $ k $ sandwiched between $ \{k\} $ and $ \{1, 1,...,1\} $ in terms of refinements, $\lambda$-choosability reveals a complex hierarchy of colourability of graphs. Assume $\lambda=\{k_1, \ldots, k_q\} $ is a partition of $ k $ and $\lambda' $ is a partition of $ k'\ge k $. We write $ \lambda\le \lambda' $ if there is a partition $\lambda''=\{k''_1, \ldots, k''_q\}$ of $k'$ with $k''_i \ge k_i$ for $i=1,2,\ldots, q$ and $\lambda'$ is a refinement of $\lambda''$. It follows from the definition that if $ \lambda\le \lambda' $, then every $\lambda$-choosable graph is $\lambda'$-choosable. It was proved in [X. Zhu, A refinement of choosability of graphs, J. Combin. Theory, Ser. B 141 (2020) 143 - 164] that the converse is also true. This paper strengthens this result and proves that for any $ \lambda\not\le \lambda' $, for any integer $g$, there exists a graph of girth at least $g$ which is $\lambda$-choosable but not $\lambda'$-choosable., Comment: 10 pages

Assume $n, m$ are positive integers and $G$ is a graph. Let $P_{n,m}$ be the graph obtained from the path with vertices $\{-m, -(m-1), \ldots, 0, \ldots, n\}$ by adding a loop at vertex $ 0$. The double cone $\Delta_{n,m}(G)$ over a graph $G$ is obtained from the direct product $G \times P_{n,m}$ by identifying $V(G) \times \{n\}$ into a single vertex $(\star, n)$, identifying $V(G) \times \{-m\}$ into a single vertex $(\star, -m)$, and adding an edge connecting $(\star, -m)$ and $(\star, n)$. This paper determines the fractional chromatic number of $\Delta_{n,m}(G)$. In particular, if $n < m$ or $n=m$ is even, then $\chi_f(\Delta_{n,m}(G)) = \chi_f(\Delta_n(G))$, where $\Delta_n(G)$ is the $n$th cone over $G$. If $n=m$ is odd, then $\chi_f(\Delta_{n,m}(G)) > \chi_f(\Delta_n(G))$. The chromatic number of $\Delta_{n,m}(G)$ is also discussed., Comment: 23 pages

A graph $G=(V,E)$ is total weight $(k,k')$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a proper $L$-total weighting, i.e., a map $\phi: V \cup E \to \mathbb{R}$ such that $\phi(z) \in L(z)$ for $z \in V \cup E$, and $\sum_{e \in E(u)}\phi(e)+\phi(u) \ne \sum_{e \in E(v)}\phi(e)+\phi(v)$ for every edge $\{u,v\}$. A graph is called nice if it contains no isolated edges. As a strengthening of the famous 1-2-3 conjecture, it was conjectured in [T. Wong and X. Zhu, Total weigt choosability of graphs, J. Graph Th. 66 (2011),198-212] that every nice graph is total weight $(1,3)$-choosable. The problem whether there is a constant $k$ such that every nice graph is total weight $(1,k)$-choosable remained open for a decade and was recently solved by Cao [L. Cao, Total weight choosability of graphs: Towards the 1-2-3 conjecture, J. Combin. Th. B, 149(2021), 109-146], who proved that every nice graph is total weight $(1, 17)$-choosable. This paper improves this result and proves that every nice graph is total weight $(1, 5)$-choosable., Comment: 29 pages, 6 figures

A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A graph $G$ is strongly fractional $r$-choosable if $G$ is $(a,b)$-choosable for all positive integers $a,b$ for which $a/b \ge r$. The strong fractional choice number of $G$ is $ch_f^s(G) = \inf \{r: G $ is strongly fractional $r$-choosable$\}$. This paper determines the strong fractional choice number of all $3$-choice critical graphs., Comment: 29 pages. arXiv admin note: text overlap with arXiv:2006.15614

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, \sigma)$ a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$. We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of M\'{a}\v{c}ajov\'{a}, Raspaud, and \v{S}koviera., Comment: 37 pages, 17 figures

Given a graph $G$, a decomposition of $G$ is a partition of its edges. A graph is $(d, h)$-decomposable if its edge set can be partitioned into a $d$-degenerate graph and a graph with maximum degree at most $h$. For $d \le 4$, we are interested in the minimum integer $h_d$ such that every planar graph is $(d,h_d)$-decomposable. It was known that $h_3 \le 4$, $h_2\le 8$, and $h_1 = \infty$. This paper proves that $h_4=1, h_3=2$, and $4 \le h_2 \le 6$., Comment: 16 pages, 5 figures

A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A characterization of $3$-choice critical graphs was given by Voigt in [On list Colourings and Choosability of Graphs, Habilitationsschrift, Tu Ilmenau(1998)]. Voigt conjectured that if $G$ is a bipartite $3$-choice critical graph, then $G$ is $(4m, 2m)$-choosable for every integer $m$. This conjecture was disproved by Meng, Puleo and Zhu in [On (4, 2)-Choosable Graphs, Journal of Graph Theory 85(2):412-428(2017)]. They showed that if $G=\Theta_{r,s,t}$ where $r,s,t$ have the same parity and $\min\{r,s,t\} \ge 3$, or $G=\Theta_{2,2,2,2p}$ with $p \ge 2$, then $G$ is bipartite $3$-choice critical, but not $(4,2)$-choosable. On the other hand, all the other bipartite 3-choice critical graphs are $(4,2)$-choosable. This paper strengthens the result of Meng, Puleo and Zhu and shows that all the other bipartite $3$-choice critical graphs are $(4m,2m)$-choosable for every integer $m$., Comment: 18 pages, 2 figures

Hedetniemi conjectured in 1966 that $\chi(G \times H) = \min\{\chi(G), \chi(H)\}$ for all graphs $G$ and $H$. Here $G\times H$ is the graph with vertex set $ V(G)\times V(H)$ defined by putting $(x,y)$ and $(x',y')$ adjacent if and only if $xx'\in E(G)$ and $yy'\in E(H)$. This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let $p$ be the minimum number of vertices in a graph of odd girth $7$ and fractional chromatic number greater than $3+4/(p-1)$. Shitov's proof shows that Hedetniemi's conjecture fails for some graphs with chromatic number about $p^22^{p+1} $ and with about $(p^22^{p+1})^{p^32^{p-1}} $ vertices. In this paper, we show that the conjecture fails already for some graphs $G$ and $H$ with chromatic number $3\lceil \frac {p+1}2 \rceil $ and with $p \lceil (p-1)/2 \rceil$ and $3 \lceil \frac {p+1}2 \rceil (p+1)-p$ vertices, respectively. The currently known upper bound for $p$ is $148$. Thus Hedetniemi's conjecture fails for some graphs $G$ and $H$ with chromatic number $225$, and with $10,952$ and $33,377$ vertices, respectively., Comment: 10 pages

We prove that if $G$ is a sparse graph --- it belongs to a fixed class of bounded expansion $\mathcal{C}$ --- and $d\in \mathbb{N}$ is fixed, then the $d$th power of $G$ can be partitioned into cliques so that contracting each of these clique to a single vertex again yields a sparse graph. This result has several graph-theoretic and algorithmic consequences for powers of sparse graphs, including bounds on their subchromatic number and efficient approximation algorithms for the chromatic number and the clique number., Comment: 14 pages

This paper disproves a conjecture of Wang, Wu, Yan and Xie, and answers in negative a question in Dvorak, Pekarek and Sereni. In return, we pose five open problems., Comment: 6 pages

A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$., Comment: 28 pages, 12 figures

Hedetniemi conjectured in 1966 that $\chi(G \times H) = \min\{\chi(G), \chi(H)\}$ for any graphs G and H. Here $G\times H$ is the graph with vertex set $ V(G)\times V(H)$ defined by putting $(x,y)$ and $(x',y')$ adjacent if and only if $xx'\in E(G)$ and $yy'\in V(H)$. This conjecture received a lot of attention in the past half century. It was disproved recently by Shitov. The Poljak-R\"{o}dl function is defined as $f(n) = \min\{\chi(G \times H): \chi(G)=\chi(H)=n\}$. Hedetniemi's conjecture is equivalent to saying $f(n)=n$ for all integer $n$. Shitov's result shows that $f(n)

This paper proves that if $G$ is a planar graph without 4-cycles and $l$-cycles for some $l\in\{5, 6, 7\}$, then there exists a matching $M$ such that $AT(G-M)\leq 3$. This implies that every planar graph without 4-cycles and $l$-cycles for some $l\in\{5, 6, 7\}$ is 1-defective 3-paintable., Comment: 16 pages, 4 figures

Let $\phi(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $\phi(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is known that $\phi(k) \le 2k+3$ and it is conjectured by Noel that $\phi(k) = 2k+3$. For a multi-set $\lambda=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a $k_{\lambda}$-list assignment $L$ for which the colour set $\cup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. Let $\phi(\lambda )$ be the minimum number of vertices in a non-$\lambda$-choosable $k_{\lambda}$-chromatic graph. Let $1_{\lambda}$ be the multiplicity of $1$ in $\lambda$, and let $o_{\lambda}$ be the number of elements in $\lambda$ that are odd integers. We prove that if $1_{\lambda} \ne k_{\lambda}$, then $2k_{\lambda}+1_{\lambda}+2 \leqslant \phi(\lambda ) \leqslant 2k_\lambda+ o_\lambda +2$. In particular, if $1_{\lambda}=o_{\lambda}=t$, i.e. $\lambda$ contains no odd integer greater than $1$, then $\phi(\lambda ) = 2k_{\lambda}+t+2$. We also prove that $\phi(\lambda) \leqslant 2k_{\lambda}+5 1_{\lambda}+3$. In particular, if $1_{\lambda}=0$, then $2k_{\lambda}+2 \leqslant \phi(\lambda) \leqslant 2k_{\lambda}+3$., Comment: 11 pages

The strong fractional choice number of a graph $G$ is the infimum of those real numbers $r$ such that $G$ is $(\lceil rm \rceil, m)$-choosable for every positive integer $m$. The strong fractional choice number of a family ${\cal G}$ of graphs is the supremum of the strong fractional choice number of graphs in ${\cal G}$. We denote by ${\cal{Q}}_k$ the class of series-parallel graphs with girth at least $k$. This paper proves that for $k=4q-1, 4q,4q+1, 4q+2$, the strong fractional number of ${\cal{Q}}_k$ is exactly $2+ \frac{1}{q}$., Comment: 7 pages

This paper proves the following result: Assume $G$ is a triangle free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\mid L(v)\mid = 4$ for each vertex $v \in V(G)-X$ and $\mid L(v)\mid = 3$ for each vertex $v \in X$, then $G$ is $L$-colourable., Comment: 19 pages, 0 figures

We prove that it is always possible to color online nonrepetitively any (partial) $k$-tree (that is, graphs with tree-width at most $k$) with $4^k$ colors. This implies that it is always possible to color online nonrepetitively cycles, trees and series-parallel graphs with $16$ colors. Our results generalize the respective (offline) nonrepetitive coloring results.

The graph coloring game is a two-player game in which, given a graph G and a set of k colors, the two players, Alice and Bob, take turns coloring properly an uncolored vertex of G, Alice having the first move. Alice wins the game if and only if all the vertices of G are eventually colored. The game chromatic number of a graph G is then defined as the smallest integer k for which Alice has a winning strategy when playing the graph coloring game on G with k colors. In this paper, we introduce and study a new version of the graph coloring game by requiring that, after each player's turn, the subgraph induced by the set of colored vertices is connected. The connected game chromatic number of a graph G is then the smallest integer k for which Alice has a winning strategy when playing the connected graph coloring game on G with k colors. We prove that the connected game chromatic number of every outerplanar graph is at most 5 and that there exist outerplanar graphs with connected game chromatic number 4. Moreover, we prove that for every integer k $\ge$ 3, there exist bipartite graphs on which Bob wins the connected coloring game with k colors, while Alice wins the connected coloring game with two colors on every bipartite graph.

We prove that $\min\{\chi(G), \chi(H)\} - \chi(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{\chi(G), \chi(H)\}/\chi(G\times H) \leq 1/2 + \epsilon$ for large values of $\min\{\chi(G), \chi(H)\}$., Comment: 5 pages

This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $\Delta(H) \le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable., Comment: 12 pages, 5 figures

This paper proves that every planar graph $G$ contains a matching $M$ such that the Alon-Tarsi number of $G-M$ is at most $4$. As a consequence, $G-M$ is $4$-paintable, and hence $G$ itself is $1$-defective $4$-paintable. This improves a result of Cushing and Kierstead [Planar Graphs are 1-relaxed, 4-choosable, {\em European Journal of Combinatorics} 31(2010),1385-1397], who proved that every planar graph is $1$-defective $4$-choosable., Comment: 11 pages

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We say $G$ is $S$-$k$-colourable if for any mapping $\sigma: E(G) \to S$ with $\sigma (x,y) = (\sigma (y,x))^{-1}$, there is a mapping $f: V(G) \to [k]=\{1,2, \ldots, k\}$ such that for each arc $e=(x,y)$, $\sigma_e(f(x)) \ne f(y)$. The concept of $S$-$k$-colouring is a common generalization of many colouring concepts, including $k$-colouring, signed $k$-colouring defined by M\'{a}\v{c}ajov\'{a}, Raspaud and \v{S}koviera, signed $k$-colouring defined by Kang and Steffen, correspondence $k$-colouring defined by Dvo\v{r}\'{a}k and Postle, and group colouring defined by Jaeger, Linial, Payan and Tarsi. We are interested in the problem as for which subset $S$ of $S_4$, every planar graph is $S$-colourable. Such a subset $S$ is called good. The famous four colour theorem is equivalent to say that $S=\{id\}$ is good. There are two conjectures on signed graph colouring, one is equivalent to $S=\{id, (12)(34)\}$ be good and the other is equivalent to $S=\{id, (12)\}$ be good. We say two subsets $S$ and $S'$ of $S_k$ are conjugate if there is a permutation $\pi \in S_k$ such that $S'= \{\pi \sigma\pi^{-1}: \sigma \in S\}$. This paper proves that if $S$ is a good subset of $S_4$ containing $id$, then $S$ is conjugate to a subset of $\{id, (12), (34), (12)(34)\}$. However, it remains an open problem if there is any good subset $S$ which contains $id$ and has cardinality $|S| \ge 2$. We also prove that $S=\{(12),(13),(23),(123),(132)\}$ is not good., Comment: 8 pages, 2 figures

Assume $k$ is a positive integer, $\lambda=\{k_1, k_2, \ldots, k_q\}$ is a partition of $k$ and $G$ is a graph. A $\lambda$-list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that the colour set $\cup_{v\in V(G)}L(v)$ can be partitioned into $q$ subsets $C_1 \cup C_2 \ldots \cup C_q$ and for each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if for each $\lambda$-list assignment $L$ of $G$, $G$ is $L$-colourable. It follows from the definition that if $\lambda =\{k\}$, then $\lambda$-choosable is the same as $k$-choosable, if $\lambda =\{1,1,\ldots, 1\}$, then $\lambda$-choosable is equivalent to $k$-colourable. For the other partitions of $k$ sandwiched between $\{k\}$ and $\{1,1,\ldots, 1\}$ in terms of refinements, $\lambda$-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions $\lambda, \lambda'$ of $k$, every $\lambda$-choosable graph is $\lambda'$-choosable if and only if $\lambda'$ is a refinement of $\lambda$. Then we concentrate on $\lambda$-choosability of planar graphs for partitions $\lambda$ of $4$. Several conjectures concerning colouring of generalized signed planar graphs are proposed and relations between these conjectures and list colouring conjectures for planar graphs are explored. In particular, it is proved that a conjecture of K\"{u}ndgen and Ramamurthi on list colouring of planar graphs is implied by the conjecture that every planar graph is $\{2,2\}$-choosable, and also implied by the conjecture of M\'{a}\v{c}ajov\'{a}, Raspaud and \v{S}koviera which asserts that every planar graph is signed MRS-$4$-colourable, and that a conjecture of Kang and Steffen asserting that every planar graph is signed KS-$4$-colourable implies that every planar graph is $\{1,1,2\}$-choosable., Comment: 10 pages, 1 figure

This paper proves that for each positive integer $m$, there is a triangle-free planar graph $G$ which is not $(3m+ \lceil \frac m{17} \rceil-1, m)$-choosable., Comment: 6 pages, 1 figure

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvo\v{r}\'{a}k and Postle. In this paper we introduce and study the fractional DP-chromatic number $\chi_{DP}^\ast(G)$. We characterize all connected graphs $G$ such that $\chi_{DP}^\ast(G) \leqslant 2$: they are precisely the graphs with no odd cycles and at most one even cycle. By a theorem of Alon, Tuza, and Voigt, the fractional list-chromatic number $\chi_\ell^\ast(G)$ of any graph $G$ equals its fractional chromatic number $\chi^\ast(G)$. This equality does not extend to fractional DP-colorings. Moreover, we show that the difference $\chi^\ast_{DP}(G) - \chi^\ast(G)$ can be arbitrarily large, and, furthermore, $\chi^\ast_{DP}(G) \geq d/(2 \ln d)$ for every graph $G$ of maximum average degree $d \geq 4$. On the other hand, we show that this asymptotic lower bound is tight for a large class of graphs that includes all bipartite graphs as well as many graphs of high girth and high chromatic number., Comment: 13 pages, 1 figure

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $\frac73n$ when $G$ is outerplanar., Comment: 15 pages, 3 figures