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We prove that a graph has an r-bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two.

A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. A hole is called {\em even} if it has an even number of vertices. An {\em even-hole-free} graph is a graph with no even holes. A vertex of a graph is {\em bisimplicial} if the set of its neighbours is the union of two cliques. In an earlier paper \cite{bisimplicial}, Addario-Berry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed, that every even-hole-free graph has a bisimplicial vertex, but we have recently been shown that the "proof" has a serious error. Here we give a proof using a different method.

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k (\mathcal{G})$ the set of all graphs that are $k$-apices of $\mathcal{G}.$ We prove that every graph in the obstruction set of $\mathcal{A}_k (\mathcal{G}),$ i.e., the minor-minimal set of graphs not belonging to $\mathcal{A}_k (\mathcal{G}),$ has size at most $2^{2^{2^{2^{\mathsf{poly}(k)}}}},$ where $\mathsf{poly}$ is a polynomial function whose degree depends on the size of the minor-obstructions of $\mathcal{G}.$ This bound drops to $2^{2^{\mathsf{poly}(k)}}$ when $\mathcal{G}$ excludes some apex graph as a minor., Comment: 48 pages and 12 figures. arXiv admin note: text overlap with arXiv:2004.12692

The proper chromatic number $\Vec{\chi}(G)$ of a graph $G$ is the minimum $k$ such that there exists an orientation of the edges of $G$ with all vertex-outdegrees at most $k$ and such that for any adjacent vertices, the outdegrees are different. Two major conjectures about the proper chromatic number are resolved. First it is shown, that $\Vec{\chi}(G)$ of any planar graph $G$ is bounded (in fact, it is at most 14). Secondly, it is shown that for every graph, $\Vec{\chi}(G)$ is at most $O(\frac{r\log r}{\log\log r})+\tfrac{1}{2}\MAD(G)$, where $r=\chi(G)$ is the usual chromatic number of the graph, and $\MAD(G)$ is the maximum average degree taken over all subgraphs of $G$. Several other related results are derived. Our proofs are based on a novel notion of fractional orientations.

Grosu [On the algebraic and topological structure of the set of Turán densities. \emph{J. Combin. Theory Ser. B} \textbf{118} (2016) 137--185] asked if there exist an integer $r\ge 3$ and a finite family of $r$-graphs whose Turán density, as a real number, has (algebraic) degree greater than~$r-1$. In this note we show that, for all integers $r\ge 3$ and $d$, there exists a finite family of $r$-graphs whose Turán density has degree at least~$d$, thus answering Grosu's question in a strong form., revised according to referees' reports

A thoroughly studied problem in Extremal Graph Theory is to find the best possible density condition in a host graph $G$ for guaranteeing the presence of a particular subgraph $H$ in $G$. One such classical result, due to Bollob\'{a}s and Thomason, and independently Koml\'{o}s and Szemer\'{e}di, states that average degree $O(k^2)$ guarantees the existence of a $K_k$-subdivision. We study two directions extending this result. On the one hand, Verstra\"ete conjectured that the quadratic bound $O(k^2)$ would guarantee already two vertex-disjoint isomorphic copies of a $K_k$-subdivision. On the other hand, Thomassen conjectured that for each $k \in \mathbb{N}$ there is some $d = d(k)$ such that every graph with average degree at least $d$ contains a balanced subdivision of $K_k$, that is, a copy of $K_k$ where the edges are replaced by paths of equal length. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on $d(k)$ remains open. In this paper, we show that the quadratic bound $O(k^2)$ suffices to force a balanced $K_k$-subdivision. This gives the optimal bound on $d(k)$ needed in Thomassen's conjecture and implies the existence of $O(1)$ many vertex-disjoint isomorphic $K_k$-subdivisions, confirming Verstra\"ete's conjecture in a strong sense., Comment: 17 pages, 4 figures

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an \emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor problems in quasi-random $k$-graphs with minimum degree $\Omega(n^{k-1})$. In particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs $H(n)$ with minimum degree $\Omega(n^2)$ and minimum codegree $\Omega(n)$ but with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite $3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense quasi-random $3$-graphs of order $n$ with minimum codegree $\Omega(n)$ and $f\mid n$ have $F$-factors. That is, $1/8$ is the density threshold for ensuring all $3$-partite $3$-graphs $F$-factors in quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$. Moreover, we show that one can not replace the minimum codegree condition by a minimum vertex degree condition. In fact, we find that for any $p\in(0,1)$ and $n\ge n_0$, there exist $p$-dense quasi-random $3$-graphs of order $n$ with minimum degree $\Omega (n^2)$ having no $K_{2,2,2}$-factor. In particular, we study the optimal density threshold of $F$-factors for each $3$-partite $3$-graph $F$ in quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$., Comment: 22 pages. Accepted by Journal of Combinatorial Theory, Series B

A graph $H$ is an induced minor of a graph $G$ if $H$ can be obtained from $G$ by vertex deletions and edge contractions. We show that there is a function $f(k, d) = O(k^{10} + 2^{d^5})$ so that if a graph has treewidth at least $f(k, d)$ and maximum degree at most $d$, then it contains a $k \times k$-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon [Eur. J. Comb., 98, 2021] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph $H$, there is a subexponential time algorithm for maximum weight independent set on $H$-induced-minor-free graphs., 7 pages. To appear in JCTB

The \emph{central levels problem} asserts that the subgraph of the $(2m+1)$-dimensional hypercube induced by all bitstrings with at least $m+1-\ell$ many 1s and at most $m+\ell$ many 1s, i.e., the vertices in the middle $2\ell$ levels, has a Hamilton cycle for any $m\geq 1$ and $1\le \ell\le m+1$.\ud This problem was raised independently by Buck and Wiedemann, Savage, Gregor and {\v{S}}krekovski, and by Shen and Williams, and it is a common generalization of the well-known \emph{middle levels problem}, namely the case $\ell=1$, and classical binary Gray codes, namely the case $\ell=m+1$.\ud In this paper we present a general constructive solution of the central levels problem.\ud Our results also imply the existence of optimal cycles through any sequence of $\ell$ consecutive levels in the $n$-dimensional hypercube for any $n\ge 1$ and $1\le \ell \le n+1$.\ud Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the $n$-dimensional hypercube, $n\geq 2$, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code.

Let $G$ be a bridgeless cubic graph. The Berge--Fulkerson Conjecture (1970s) states that $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan--Raspaud Conjecture (1994), which states that $G$ admits three perfect matchings such that every edge of $G$ belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that $G$ admits two perfect matchings whose deletion yields a bipartite subgraph of $G$. It can be shown that given an arbitrary perfect matching of $G$, it is not always possible to extend it to a list of three or six perfect matchings satisfying the statements of the Fan--Raspaud and the Berge--Fulkerson conjectures, respectively. In this paper, we show that given any $1^+$-factor $F$ (a spanning subgraph of $G$ such that its vertices have degree at least 1) and an arbitrary edge $e$ of $G$, there always exists a perfect matching $M$ of $G$ containing $e$ such that $G\setminus (F\cup M)$ is bipartite. Our result implies Mazzuoccolo's conjecture, but not only. It also implies that given any collection of disjoint odd circuits in $G$, there exists a perfect matching of $G$ containing at least one edge of each circuit in this collection., 13 pages, 8 figures

It is known that $A$-paths of length $0$ mod $m$ satisfy the Erd\H{o}s-P\'osa property if $m=2$ or $m=4$, but not if $m > 4$ is composite. We show that if $p$ is prime, then $A$-paths of length $0$ mod $p$ satisfy the Erd\H{o}s-P\'osa property. More generally, in the framework of undirected group-labelled graphs, we characterize the abelian groups $\Gamma$ and elements $\ell \in \Gamma$ for which the Erd\H{o}s-P\'osa property holds for $A$-paths of weight $\ell$., Comment: arXiv admin note: text overlap with arXiv:2009.11266

A graph $H$ is ubiquitous if for every graph $G$ that for every natural number $n$ contains $n$ vertex-disjoint $H$-minors contains infinitely many vertex-disjoint $H$-minors. Andreae conjectured that every locally finite graph is ubiquitous. We give a disconnected counterexample to this conjecture. It remains open whether every connected locally finite graph is ubiquitous.

Let $G$ be a $K_p$-free graph. We say $e$ is a $K_p$-saturating edge of $G$ if $e\notin E(G)$ and $G+e$ contains a copy of $K_p$. Denote by $f_p(n, e)$ the minimum number of $K_p$-saturating edges that an $n$-vertex $K_p$-free graph with $e$ edges can have. Erd\H{o}s and Tuza conjectured that $f_4(n,\lfloor n^2/4\rfloor+1)=\left(1 + o(1)\right)\frac{n^2}{16}.$ Balogh and Liu disproved this by showing $f_4(n,\lfloor n^2/4\rfloor+1)=(1+o(1))\frac{2n^2}{33}$. They believed that a natural generalization of their construction for $K_p$-free graph should also be optimal and made a conjecture that $f_{p+1}(n,ex(n,K_p)+1)=\left(\frac{2(p-2)^2}{p(4p^2-11p+8)}+o(1)\right)n^2$ for all integers $p\ge 3$. The main result of this paper is to confirm the above conjecture of Balogh and Liu.

The feasible region $\Omega_{{\rm ind}}(F)$ of a graph $F$ is the collection of points $(x,y)$ in the unit square such that there exists a sequence of graphs whose edge densities approach $x$ and whose induced $F$-densities approach $y$. A complete description of $\Omega_{{\rm ind}}(F)$ is not known for any $F$ with at least four vertices that is not a clique or an independent set. The feasible region provides a lot of combinatorial information about $F$. For example, the supremum of $y$ over all $(x,y)\in \Omega_{{\rm ind}}(F)$ is the inducibility of $F$ and $\Omega_{{\rm ind}}(K_r)$ yields the Kruskal-Katona and clique density theorems. We begin a systematic study of $\Omega_{{\rm ind}}(F)$ by proving some general statements about the shape of $\Omega_{{\rm ind}}(F)$ and giving results for some specific graphs $F$. Many of our theorems apply to the more general setting of quantum graphs. For example, we prove a bound for quantum graphs that generalizes an old result of Bollob\'as for the number of cliques in a graph with given edge density. We also consider the problems of determining $\Omega_{{\rm ind}}(F)$ when $F=K_r^-$, $F$ is a star, or $F$ is a complete bipartite graph. In the case of $K_r^-$ our results sharpen those predicted by the edge-statistics conjecture of Alon et. al. while also extending a theorem of Hirst for $K_4^-$ that was proved using computer aided techniques and flag algebras. The case of the 4-cycle seems particularly interesting and we conjecture that $\Omega_{{\rm ind}}(C_4)$ is determined by the solution to the triangle density problem, which has been solved by Razborov., Comment: revised according to two referee reports

An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $\Delta=4$ or $\Delta\ge 6$, every connected $n$-vertex graph of maximum degree at most $\Delta$ has an independent dominating set of size at most $(1-\frac{\Delta}{\lfloor\Delta^2/4\rfloor+\Delta})(n-1)+1$. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most $(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta})n$., Comment: 14 pages

In 1943, Hadwiger conjectured that every graph with no K t + 1 minor is t-colorable for every t ≥ 0 . While Hadwiger's conjecture does not hold for list-coloring, the linear weakening is conjectured to be true. In the 1980 s, Kostochka and Thomason independently proved that every graph with no K t minor has average degree O ( t log ⁡ t ) and thus is O ( t log ⁡ t ) -list-colorable. Recently, the authors and Song proved that every graph with no K t minor is O ( t ( log ⁡ t ) β ) -colorable for every β > 1 4 . Here, we build on that result to show that every graph with no K t minor is O ( t ( log ⁡ t ) β ) -list-colorable for every β > 1 4 . Our main new tool is an upper bound on the number of vertices in highly connected K t -minor-free graphs: We prove that for every β > 1 4 , every Ω ( t ( log ⁡ t ) β ) -connected graph with no K t minor has O ( t ( log ⁡ t ) 7 / 4 ) vertices.

For a graph G and a set Z of four distinct vertices of G, a diamond on Z is a subgraph of G such that, for some labeling Z = { v 1 , v 2 , v 3 , v 4 } , there are three internally disjoint paths P 1 , P 2 , P 3 with end vertices v 1 , v 2 with v 3 , v 4 on P 1 , P 2 , respectively. Therefore, this yields a K 4 − -subdivision with branch vertices on Z. We characterize graphs G that contain no diamond on a prescribed set Z of four vertices, under the assumption that for every v ∈ Z there are three paths of G from v to Z − { v } , mutually disjoint except for v. Moreover, we can find two “different” such subdivisions, if one exists. Our proof is based on Mader's S-paths theorem.

We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of checking that a hypergraph $\mathcal H$ with large minimum degree that omits the forbidden structures is vertex-extendable. This means that if $v$ is a vertex of $\mathcal H$ and ${\mathcal H} -v$ is a subgraph of the extremal configuration(s), then $\mathcal H$ is also a subgraph of the extremal configuration(s). In many cases vertex-extendability is quite easy to verify. We illustrate our approach by giving new short proofs of hypergraph stability results of Pikhurko, Hefetz-Keevash, Brandt-Irwin-Jiang, Bene Watts-Norin-Yepremyan and others. Since our method always yields minimum degree stability, which is the strongest form of stability, in some of these cases our stability results are stronger than what was known earlier. Along the way, we clarify the different notions of stability that have been previously studied., revised according to two referee reports

A basic result in graph theory says that any n-vertex tournament with in- and out-degrees larger than [Formula presented] contains a Hamilton cycle, and this is tight. In 1990, Bollobás and Häggkvist significantly extended this by showing that for any fixed k and ε>0, and sufficiently large n, all tournaments with degrees at least [Formula presented] contain the k-th power of a Hamilton cycle. Up until now, there has not been any progress on determining a more accurate error term in the degree condition, neither in understanding how large n should be in the Bollobás-Häggkvist theorem. We essentially resolve both of these questions. First, we show that if the degrees are at least [Formula presented] for some constant c=c(k), then the tournament contains the k-th power of a Hamilton cycle. In particular, in order to guarantee the square of a Hamilton cycle, one only needs a constant additive term. We also present a construction which, modulo a well known conjecture on Turán numbers for complete bipartite graphs, shows that the error term must be of order at least n1−1/⌈(k−1)/2⌉, which matches our upper bound for all even k. For odd k, we believe that the lower bound can be improved. Indeed, we show that for k=3, there are tournaments with degrees [Formula presented] and no cube of a Hamilton cycle. In addition, our results imply that the Bollobás-Häggkvist theorem already holds for n=ε−Θ(k), which is best possible., Journal of Combinatorial Theory. Series B, 158, ISSN:0095-8956

The number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2 n / 17700000 . On the other hand, for infinitely many n, there exists a triangle-free planar graph with separating cycles of length 4 and 5 whose number of proper vertex-3-colorings is 2 15 n / log 2 ⁡ ( n ) .

A $k$-connected set in an infinite graph, where $k > 0$ is an integer, is a set of vertices such that any two of its subsets of the same size $\ell \leq k$ can be connected by $\ell$ disjoint paths in the whole graph. We characterise the existence of $k$-connected sets of arbitrary but fixed infinite cardinality via the existence of certain minors and topological minors. We also prove a duality theorem for the existence of such $k$-connected sets: if a graph contains no such $k$-connected set, then it has a tree-decomposition which, whenever it exists, precludes the existence of such a $k$-connected set., 50 pages, 8 figures

Let $T_G(x,y)$ be the Tutte polynomial of a graph $G$. In this paper we show that if $(G_n)_n$ is a sequence of $d$-regular graphs with girth $g(G_n)\to \infty$, then for $x\geq 1$ and $0\leq y\leq 1$ we have $$\lim_{n\to \infty}T_{G_n}(x,y)^{1/v(G_n)}=t_d(x,y),$$ where $$t_d(x,y)=\left\{\begin{array}{lc} (d-1)\left(\frac{(d-1)^2}{(d-1)^2-x}\right)^{d/2-1}&\ \ \mbox{if}\ x\leq d-1,\\ x\left(1+\frac{1}{x-1}\right)^{d/2-1} &\ \ \mbox{if}\ x> d-1. \end{array}\right.$$ independently of $y$ if $0\leq y\leq 1$. If $(G_n)_n$ is a sequence of random $d$-regular graphs, then the same statement holds true asymptotically almost surely. This theorem generalizes results of McKay ($x=1,y=1$, spanning trees of random $d$-regular graphs) and Lyons ($x=1,y=1$, spanning trees of large-girth $d$-regular graphs). Interesting special cases are $T_G(2,1)$ counting the number of spanning forests, $T_G(2,0)$ counting the number of acyclic orientations.

One of the major problems in combinatorics is to determine the number of $r$-uniform hypergraphs ($r$-graphs) on $n$ vertices which are free of certain forbidden structures. This problem dates back to the work of Erd\H{o}s, Kleitman and Rothschild, who showed that the number of $K_r$-free graphs on $n$ vertices is $2^{\text{ex}(n,K_r)+o(n^2)}$. Their work was later extended to forbidding graphs as induced subgraphs by Pr\"omel and Steger. Here, we consider one of the most basic counting problems for $3$-graphs. Let $E_1$ be the $3$-graph with $4$ vertices and $1$ edge. What is the number of induced $\{K_4^3,E_1\}$-free $3$-graphs on $n$ vertices? We show that the number of such $3$-graphs is of order $n^{\Theta(n^2)}$. More generally, we determine asymptotically the number of induced $\mathcal{F}$-free $3$-graphs on $n$ vertices for all families $\mathcal{F}$ of $3$-graphs on $4$ vertices. We also provide upper bounds on the number of $r$-graphs on $n$ vertices which do not induce $i \in L$ edges on any set of $k$ vertices, where $L \subseteq \big \{0,1,\ldots,\binom{k}{r} \big\}$ is a list which does not contain $3$ consecutive integers in its complement. Our bounds are best possible up to a constant multiplicative factor in the exponent when $k = r+1$. The main tool behind our proof is counting the solutions of a constraint satisfaction problem., Comment: 18 pages

A necklace can be considered as a cyclic list of n red and n blue beads in an arbitrary order. In the necklace folding problem the goal is to find a large crossing-free matching of pairs of beads of different colors in such a way that there exists a “folding” of the necklace, that is a partition into two contiguous arcs, which splits the beads of any matching edge into different arcs. We give counterexamples for some conjectures about the necklace folding problem, also known as the separated matching problem. The main conjecture (given independently by three sets of authors) states that [Formula presented], where μ is the ratio of the maximum number of matched beads to the total number of beads. We refute this conjecture by giving a construction which proves that μ≤2−2

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $[n]:=\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $m/n\leq 1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values. In contrast, this is not true anymore in the dense regime, when $m/n>1$, where the maximum degree of $P(n,m)$ is not concentrated on any subset of $[n]$ with bounded size., arXiv admin note: substantial text overlap with arXiv:2010.15083

Over recent years there has been much interest in both Tur\'an and Ramsey properties of vertex ordered graphs. In this paper we initiate the study of embedding spanning structures into vertex ordered graphs. In particular, we introduce a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect $H$-tiling in an ordered graph. In the (unordered) graph setting, this problem was resolved by K\"uhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009]. We use our general framework to resolve the perfect $H$-tiling problem for all ordered graphs $H$ of interval chromatic number $2$. Already in this restricted setting the class of extremal examples is richer than in the unordered graph problem. In the process of proving our results, novel approaches to both the regularity and absorbing methods are developed., Comment: 23 pages, author accepted manuscript to appear in JCTB

V práci studujeme efekt hranové eliminace na cyklickou souvislost kubického grafu. Dokážeme, že kromě tří výjimečných grafů, v kubickom grafe existuje hrana, jejíž eliminace zmenší cyklickou souvislost nejvýše o jednotku. Zvláštní chování kubických grafů souvislosti 6 vzhledem k eliminaci hran nás přivedlo k charakterizaci Issacsových grafů, které hrají důležitou roli při studiu kubických grafů. Hlavní výsledek je použitý na důkaz tvrzení, že pro každý 5-souvislý kubický graf existuje rozklad vrcholové množiny na indukovaný strom a indukovaný podgraf s nejvýše jednou hranou. Tento výsledek zobecňuje větu Payana a Sakharoviche z roku 1975. Edge-elimination is an operation of removing an edge of a cubic graph together with its endvertices and suppressing the resulting 2-valent vertices. We study the effect of this operation on the cyclic connectivity of a cubic graph. Disregarding a small number of cubic graphs with no more than six vertices, this operation cannot decrease cyclic connectivity by more than two. We show that apart from three exceptional graphs (the cube, the twisted cube, and the Petersen graph) every 2-connected cubic graph on at least eight vertices contains an edge whose elimination decreases cyclic connectivity by at most one. The proof reveals an unexpected behaviour of connectivity 6, which requires a detailed structural analysis featuring the Isaacs flower snarks and their natural generalisation, the twisted Isaacs graphs, as forced structures. A complete characterisation of this family, which includes the Heawood graph as a sporadic case, serves as the main tool for excluding the existence of exceptional graphs in connectivity 6. As an application we show that every cyclically 5-edge-connected cubic graph has a decycling set of vertices whose removal leaves a tree and the set itself has at most one edge between its vertices. This strengthens a classical result of Payan and Sakarovitch (1975) .

Hadwiger's conjecture states that every Kt-minor free graph is (t−1)-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd Kt-minor is (t−1)-colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form Ct for some constant C>0 exists. We show that if every graph without a Kt-minor is f(t)-colorable, then every graph without an odd Kt-minor is 2f(t)-colorable. As a direct corollary, we present a new state of the art bound O(tloglogt) for the chromatic number of graphs with no odd Kt-minor. Moreover, the short proof of our result substantially simplifies the proofs of all the previous asymptotic bounds for the chromatic number of odd Kt-minor free graphs established in a sequence of papers during the last 12 years., Journal of Combinatorial Theory. Series B, 155, ISSN:0095-8956