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In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, where one induces a graph with maximum degree two, and the remaining two are independent sets.

An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $q$-coloring of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r$ and the sizes of any two color classes differ by at most $1.$ Let the \emph{strong equitable vertex $r$-arboricity} of a graph $G,$ denoted by $va^\equiv_r (G)$, be the minimum $p$ such that $G$ has an equitable $(q, r)$-tree-coloring for every $q\geq p.$ The values of $va^\equiv_1 (K_{n,n})$ were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of $va^\equiv_1 (K_{n,n})$ were found in some special cases. In this paper, we extend their results by giving the exact values of $va^\equiv_1 (K_{n,n})$ for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each $va^\equiv_1 (K_{m,n})$ and $va^\equiv_1 (G)$ where $G$ is a balanced complete $k$-partite graph $K_{n,\ldots,n}.$

In this work, we introduce DPG-coloring using the concepts of DP-coloring and variable degeneracy to modify the proofs on the following papers: (i) DP-3-coloring of planar graphs without $4$, $9$-cycles and cycles of two lengths from $\{6, 7, 8\}$ (R. Liu, S. Loeb, M. Rolek, Y. Yin, G. Yu, Graphs and Combinatorics 35(3) (2019) 695-705), (ii) Every planar graph without $i$-cycles adjacent simultaneously to $j$-cycles and $k$-cycles is DP-$4$-colorable when $\{i, j, k\}=\{3, 4, 5\}$ (P. Sittitrai, K. Nakprasit, arXiv:1801.06760(2019) preprint), (iii) Every planar graph is $5$-choosable (C. Thomassen, J. Combin. Theory Ser. B 62 (1994) 180-181). Using this modification, we obtain more results on list coloring, DP-coloring, list-forested coloring, and variable degeneracy., Comment: arXiv admin note: text overlap with arXiv:1807.00815

Xu and Wu proved that if every 5-cycle of a planar graph G is not simultaneously adjacent to 3-cycles and 4-cycles, then G is 4-choosable. In this paper, we improve this result as follows. If G is a planar graph without pairwise adjacent 3-,4-,5-, and 6-cycle, then G is 4-choosable., Comment: 17 pages and 2 figures

In \cite{listnoC3adjC4}), Borodin and Ivanova proved that every planar graph without $4$-cycles adjacent to $3$-cycle is list vertex-$2$-aborable. In fact, they proved a more general result. Inspired by these results and DP-coloring which becomes a widely studied topic, we introduce a generalization on variable degeneracy including list vertex arboricity. We use this notion to extend a general result by Borodin and Ivanova. Not only that this theorem implies results about planar graphs without $4$-cycles adjacent to $3$-cycle by Borodin and Ivanova, it implies many other results including a result by Kim and Yu \cite{KimY} that every planar graph without $4$-cycles adjacent to $3$-cycle is DP-$4$-colorable., Comment: 8 pages, 5 figures

It is known that DP-coloring is a generalization of a list coloring in simple graphs and many results in list coloring can be generalized in those of DP-coloring. In this work, we introduce a relaxed DP-coloring which is a generalization if a relaxed list coloring. We also shows that every planar graph $G$ without $4$-cycles or $6$-cycles is DP-$(k,d)^*$-colorable. It follows immediately that $G$ is $(k,d)^*$-choosable., Comment: arXiv admin note: substantial text overlap with arXiv:1801.06760

DP-coloring is a generalization of a list coloring in simple graphs. Many results in list coloring can be generalized in those of DP-coloring. Kim and Ozeki showed that planar graphs without $k$-cycles where $k=3,4,5,$ or $6$ are DP-$4$-colorable. Recently, Kim and Yu extended the result on $3$- and $4$-cycles by showing that planar graphs without triangles adjacent to $4$-cycles are DP-$4$-colorable. Xu and Wu showed that planar graphs without $5$-cycles adjacent simultaneously to $3$-cycles and $4$-cycles are $4$-choosable. In this paper, we extend the result on $5$-cycles and triangles adjacent to $4$-cycles by showing that planar graphs without $i$-cycles adjacent simultaneously to $j$-cycles and $k$-cycles are DP-$4$-colorable when $\{i,j,k\}=\{3,4,5\}.$ This also generalizes the result of Xu and Wu.

Xu and Wu proved that if every $5$-cycle of a planar graph $G$ is not simultaneously adjacent to $3$-cycles and $4$-cycles, then $G$ is $4$-choosable. In this paper, we improve this result as follows. Let $\{i, j, k, l\} = \{3,4,5,6\}.$ For any chosen $i,$ if every $i$-cycle of a planar graph $G$ is not simultaneously adjacent to $j$-cycles, $k$-cycles, and $l$-cycles, then $G$ is $4$-choosable.

Consider two conditions on a graph: (1) each 5-cycle is not a subgraph of 5-wheel and does not share exactly one edge with 3-cycle, and (2) each 5-cycle is not adjacent to two 3-cycles and is not adjacent to a 4-cycle with chord. We show that if a planar graph $G$ satisfies one of the these conditions, then $G$ is 4-choosable. This yields that if each 5-cycle of a planar graph $G$ is not adjacent a 3-cycle, then $G$ is 4-choosable.

Let $G$ be a graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for each positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for each positive integer $k.$ Finally, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4), (3,5),$ and $(2,9).$

Let ${\rm col_g}(G)$ be the game coloring number of a given graph $G.$ Define the game coloring number of a family of graphs $\mathcal{H}$ as ${\rm col_g}(\mathcal{H}) := \max\{{\rm col_g}(G):G \in \mathcal{H}\}.$ Let $\mathcal{P}_k$ be the family of planar graphs of girth at least $k.$ We show that ${\rm col_g}(\mathcal{P}_7) \leq 5.$ This result extends a result about the coloring number by Wang and Zhang {WZ11} (${\rm col_g}(\mathcal{P}_8) \leq 5).$ We also show that these bounds are sharp by constructing a graph $G$ where $G \in {\rm col_g}(\mathcal{P}_k) \geq 5$ for each $k \leq 8$ such that ${\rm col_g}(G)=5.$ As a consequence, ${\rm col_g}(\mathcal{P}_k) = 5$ for $k =7,8.$

A $(q,t)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $t.$ A $(q,\infty)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest. Wu, Zhang, and Li introduced the concept of \emph{equitable $(q, t)$-tree-coloring} (respectively, \emph{equitable $(q, \infty)$-tree-coloring}) which is a $(q,t)$-tree-coloring (respectively, $(q, \infty)$-tree-coloring) such that the sizes of any two color classes differ by at most one. Among other results, they obtained a sharp upper bound on the minimum $p$ such that $K_{n,n}$ has an equitable $(q, 1)$-tree-coloring for every $q\geq p.$ In this paper, we obtain a polynomial time criterion to decide if a complete bipartite graph has an equitable $(q,t)$-tree-coloring or an equitable $(q,\infty)$-tree-coloring. Nevertheless, deciding if a graph $G$ in general has an equitable $(q,t)$-tree-coloring or an equitable $(q,\infty)$-tree-coloring is NP-complete., Comment: arXiv admin note: text overlap with arXiv:1506.03913

A $q$-\emph{equitable coloring} of a graph $G$ is a proper $q$-coloring such that the sizes of any two color classes differ by at most one. In contrast with ordinary coloring, a graph may have an equitable $q$-coloring but has no equitable $(q+1)$-coloring. The \emph{equitable chromatic threshold} is the minimum $p$ such that $G$ has an equitable $q$-coloring for every $q\geq p.$ In this paper, we establish the notion of $p(q: n_1,\ldots, n_k)$ which can be computed in linear-time and prove the following. Assume that $K_{n_1,\ldots,n_k}$ has an equitable $q$-coloring. Then $p(q: n_1,\ldots, n_k)$ is the minimum $p$ such that $K_{n_1,\ldots,n_k}$ has an equitable $r$-coloring for each $r$ satisfying $p \leq r \leq q.$ Since $K_{n_1,\ldots,n_k}$ has an equitable $(n_1+\cdots+n_k)$-coloring, the equitable chromatic threshold of $K_{n_1,\ldots,n_k}$ is $p(n_1+\cdots+n_k: n_1,\ldots, n_k).$ We find out later that the aforementioned immediate consequence is exactly the same as the formula of Yan and Wang \cite{YW12}. Nonetheless, the notion of $p(q: n_1,\ldots, n_k)$ can be used for each $q$ in which $K_{n_1,\ldots,n_k}$ has an equitable $q$-coloring and the proof presented here is much shorter., Comment: arXiv admin note: text overlap with arXiv:1506.03913

A $(q,r)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r.$ An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $(q,r)$-tree-coloring such that the sizes of any two color classes differ by at most one. Let the \emph{strong equitable vertex $r$-arboricity} be the minimum $p$ such that $G$ has an equitable $(q, r)$-tree-coloring for every $q\geq p.$ In this paper, we find the exact value for each $va^\equiv_2(K_{m,n})$ and $va^\equiv_2(K_{l,m,n}).$

The online list coloring is a widely studied topic in graph theory. A graph $G$ is 2-paintable if we always have a strategy to complete a coloring in an online list coloring of $G$ in which each vertex has a color list of size 2. In this paper, we focus on the online list coloring game in which the number of colors is known in advance. We say that $G$ is $[2,t]$-paintable if we always have a strategy to complete a coloring in an online list coloring of $G$ in which we know that there are exactly $t$ colors in advance, and each vertex has a color list of size 2. Let $M(G)$ denote the maximum $t$ in which $G$ is not $[2,t]$-paintable, and $m(G)$ denote the minimum $t \geq 2$ in which $G$ is not $[2,t]$-paintable. We show that if $G$ is not 2-paintable, then $2 \leq m(G) \leq 4,$ and $n \leq M(G) \leq 2n-3.$ Furthermore, we characterize $G$ with $m(G)\in \{2,3,4\}$ and $M(G) \in \{n, n+1, 2n-3\},$ respectively.

An \emph{equitable coloring} of a graph is a proper vertex coloring such that the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree $\Delta \geq 2$ has an equitable coloring with $\Delta$ colors, except when $G$ is a complete graph or an odd cycle or $\Delta$ is odd and $G=K_{\Delta,\Delta}.$ Nakprasit proved the conjecture holds for planar graphs with maximum degree at least 9. Zhu and Bu proved that the conjecture holds for every $C_3$-free planar graph with maximum degree at least 8 and for every planar graph without $C_4$ and $C_5$ with maximum degree at least 7. In this paper, we prove that the conjecture holds for planar graphs in various settings, especially for every $C_3$-free planar graph with maximum degree at least 6 and for every planar graph without $C_4$ with maximum degree at least 7, which improve or generalize results on equitable coloring by Zhu and Bu. Moreover, we prove that the conjecture holds for every planar graph of girth at least 6 with maximum degree at least 5.