Sergey L. Berlov, Ilya I. Bogdanov
In this paper, we present the lower bounds for the number of vertices in a graph with a large chromatic number containing no small odd cycles.
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Two vertices of an odd-distance graph are connected by an edge if and only if their Euclidean distance is an odd integer. We construct a 6-chromatic odd-distance graph in the plane.
The odd chromatic number and the conflict-free chromatic number are new graph parameters introduced by Petruševski and Škrekovski [2021] and Fabrici, Lužar, Rindošová and Soták [2022] respectively. In this note, we show that graphs with bounded $2$-strong colouring number have bounded odd chromatic number and bounded conflict-free chromatic number. This implies that graph classes with bounded expansion have bounded odd chromatic number and bounded conflict-free chromatic number. Moreover, it follows by known results that the odd chromatic number and the conflict-free chromatic number of $k$-pl…
We prove a conjecture of Andras Gyarfas, that for all k,t, every graph with clique number at most k and sufficiently large chromatic number has an odd hole of length at least t.
Upper bounds on the odd graceful chromatic number of graphs
本文研究二部图的奇优美色数，给出若干仅依赖顶点数或其二部划分相关图色数的新上界。
An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyarfas made the conjecture that for all t there exists n such that every graph with no K_t subgraph and no odd hole is n-colourable. We prove this conjecture.
On a conjecture about the strong odd chromatic number of planar graphs
本文研究了平面图的强奇数染色数，通过构造反例验证了一个关于其上界猜想的假设。
We investigate group-theoretic "signatures" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which …
Resolving a conjecture of Bollobás and Erdős, Gyárfás proved that every graph $G$ of chromatic number $k+1\geq 3$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ distinct odd lengths. We strengthen this prominent result by showing that such $G$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ consecutive odd lengths. Along the way, combining extremal and structural tools, we prove a stronger statement that every graph of chromatic number $k+1\geq 7$ contains $k$ cycles of consecutive lengths, except that some block is $K_{k+1}$. As corollaries, this confirms a conjecture of Verstraëte and answers…
Petruševski and Škrekovski \cite{odd9} recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph $G$ is said to be \emph{odd} if for each non-isolated vertex $x \in V(G)$ there exists a colour $c$ appearing an odd number of times in $N(x)$. Petruševski and Škrekovski proved that for any planar graph $G$ there is an odd colouring using at most $9$ colours and, together with Caro \cite{oddremarks}, showed that $8$ colours are enough for a significant family of planar graphs. We show that $8$ colours suffice for all planar graphs.
A class of simple graphs such as ${\cal G}$ is said to be {\it odd-girth-closed} if for any positive integer $g$ there exists a graph $G \in {\cal G}$ such that the odd-girth of $G$ is greater than or equal to $g$. An odd-girth-closed class of graphs ${\cal G}$ is said to be {\it odd-pentagonal} if there exists a positive integer $g^*$ depending on ${\cal G}$ such that any graph $G \in {\cal G}$ whose odd-girth is greater than $g^*$ admits a homomorphism to the five cycle (i.e. is $C_{_{5}}$-colorable). In this article, we show that finding the odd girth of generalized Petersen graphs can be t…
Give a digraph $D=(V(D),A(D))$, let $\partial^+_D(v)=\{vw|w\in N^+_D(v)\}$ and $\partial^-_D(v)=\{uv|u\in N^-_D(v)\}$ be semi-cuts of $v$. A mapping $φ:A(D)\rightarrow [k]$ is called a weak-odd $k$-edge coloring of $D$ if it satisfies the condition: for each $v\in V(D)$, there is at least one color with an odd number of occurrences on each non-empty semi-cut of $v$. We call the minimum integer $k$ the weak-odd chromatic index of $D$. When limit to 2 colors, use $def(D)$ to denote the defect of $D$, the minimum number of vertices in $D$ at which the above condition is not satisfied. In this pap…
Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log t})$-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no $K_t$ minor, we make the first improvement on this bound by showing that every graph with no odd $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$.
Upper Bounds of the Odd Chromatic Number of a Graph in terms of its Thickness
本文研究图的奇色数上界，证明最小度至少为2倍厚度减1且围长至少为6的图可被6倍厚度种颜色奇染色。
It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has at least one color that appears an odd number of times in its neighborhood. The smallest number of colors that admits an odd coloring of a graph $ G $ is denoted by $χ_{o}(G)$. In this paper, we prove that if $G$ is tortoidal, then $χ_{o}\left({G}\right)\le9$; Note that $K_7$ is a toroidal graph, the upper bound is no less than $7$.
We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$ satisfies the list-edge-coloring conjecture. When $Δ(G) \geq 4$, we in fact prove a stronger result about kernel-perfect orientations in $L(G)$ which implies that $G$ is $(mΔ(G):m)$-edge-choosable and $Δ(G)$-edge-paintable for every $m \geq 1$.
Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1) Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; (2) For all $l$, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than $l$; (3) For all $k$, $l$, every graph with no clique of size more than $k$ and sufficiently large chroma…
For a set H of tournaments, we say H is heroic if every tournament, not containing any member of H as a subtournament, has bounded chromatic number. Berger et al. explicitly characterized all heroic sets containing one tournament. Motivated by this result, we study heroic sets containing two tournaments. We give a necessary condition for a set containing two tournaments to be heroic. We also construct infinitely many minimal heroic sets of size two.
A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an odd colouring of G. Scott [Graphs and Combinatorics, 2001] proved that a graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph G is k-odd colourable if it can be partitioned into at most k odd induced subgraphs. We initiate the systematic study of odd colouring and odd chromatic n…
In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and their edges are pairs of vectors at a distance from a given set of postive numbers apart. A series of new lower bounds are obtained for the chromatic numbers of such graphs with different restrictions on the clique numbers and the girths.
Here we prove that if G has independence no. 2 and clique size omega with omega less than or equal to 11, then (1) chromatic no. is less than or equal to (omega2+12omega-13)/8, if omega is odd, and (2) chromatic no. is less than or equal to (omega2+10omega)/8, if omega is even. We further conjecture that the results are true in general for all omega. We also conjecture that (A) if omega is odd and R(3, omega) is even, then R(3, omega) = (omega2+8omega-9)/4, (B) if omega and R(3, omega) are both odd, then (omega2+8omega-13)/4, (C) if omega and R(3, omega) are both even, then R(3, omega) = (omeg…
Clustered Colouring of Odd-$H$-Minor-Free Graphs
本文研究排除奇子式图类的聚类色数，通过改编Norin等人的证明方法，证明该色数与排除图H的树深密切相关。
Cliques and High Odd Holes in Graphs with Chromatic Number Equal to Maximum Degree
本文为满足色数等于最大度且非$\overline{C_7}$的连通图提供了一个统一自包含的证明，证明了其必然包含特定团或高奇洞，避免了先前研究中依赖强完美图定理的繁琐分类。
Maximum odd induced subgraph of a graph concerning its chromatic number
本文研究图的奇诱导子图最大阶数问题，证明 Scott 猜想对二部图不成立但对线图成立，并否定 Berman 等人的相关猜想。
For a prime power $q$, let $ER_q$ denote the Erdős-Rényi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $χ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best possible up to a constant factor of at most 2. If $q$ is an odd power of an odd prime and satisfies some condition on irreducible polynomials, then we improve the best known upper bound for $χ(ER_{q})$ substantially. We also show that for sufficiently large $q$, every $ER_q$ contains a subgraph that is not 3-chromatic and has at most 36 vertices.
There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conje…
A hole is an induced cycle of length at least 4. A graph is called a pentagraph if it has no cycles of length 3 or 4 and has no holes of odd length at least 7, and is called a heptagraph if it has no cycles of length less than 7 and has no holes of odd length at least 9. Let $ł\ge 2$ be an integer. The current authors proved that a graph is 4- colorable if it has no cycles of length less than $2ł+1$ and has no holes of odd length at least $2ł+3$. Confirming a conjecture of Plummer and Zha, Chudnovsky and Seymour proved that every pentagraph is 3-colorable. Following their idea, we show that ev…
It is proved that if $G$ is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of $G$ has no roots in the interval $(1,t_1]$, where $t_1 \approx 1.2904$ is the smallest real root of the polynomial $(t-2)^6 +4(t-1)^2(t-2)^3 -(t-1)^4$. We also construct a family of graphs containing such spanning trees with chromatic roots converging to $t_1$ from above. We employ the Whitney $2$-switch operation to manage the analysis of an infinite class of chromatic polynomials.
A proper vertex coloring $φ$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $φ^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of $G$, denoted $χ_o(G)$, is the odd chromatic number. Odd colorings were recently introduced in [M.~Petruševski, R.~Škrekovski: \textit{Colorings with neighborhood parity condition}]. Here we discuss various basic properties of this new graph parameter, characterize acyclic graphs and hypercubes in terms of odd chromatic number, establish several upper bounds in regard to d…
Alternating odd cycles and orientations of Kneser-like graphs
本文研究特定图的定向问题，聚焦拓扑χ-染色图，探讨其是否存在避免交错奇环的定向或使所有最短奇环为交错奇环的定向，并针对Kneser图等多个图族给出结果。
A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. A full house is a graph composed by a vertex adjacent to both ends of an edge in $K_4$ . Let $H$ be the complement of a cycle on 7 vertices. Chudnovsky et al [6] proved that every (odd hole, $K_4$)-free graph is 4-colorable and is 3-colorable if it does not has $H$ as an induced subgraph. In this paper, we use the proving technique of Chudnovsky et al to generalize this conclusion to (odd hole, full house)-free graphs, and prove that for (odd hole, full house)-free graph $G$, $χ(G)\le ω(G)+1$, and the equ…
An Improved Upper Bound for the Strong Odd Chromatic Number of Planar Graphs
本文将平面图强奇数色数的上界从388改进至368，还构造了强奇数色数为20的平面图，回应了相关研究问题。
Hypergraph Universality and Erdos-Posa Failure for Chromatically Rich Odd Cycles
本文证明对任意k≥3，任意有限丛可由k-坏奇圈的极小终端迹实现，保持打包-覆盖数，导致有限拥塞和分数Erdős-Pósa性质失效，与k=2情形形成对比。
On the Turán number of odd-ballooning of $3$-chromatic graphs
本文应用 Simonovits 的渐进归纳法，确定了当每个替换奇环长度至少为5时，一类 $3$-色图的奇气球化 Turán 数，并推论出奇轮图、扇图、书图和友谊图等著名图类的 Turán 数。
The perfect divisibility and chromatic number of some odd hole-free graphs
本文研究了无奇环、锤子和K_{2,3}的图的完美可分性，并给出了短孔图在不同约束下的色数上界。
We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $χ(G) \leq χ_\ell(G) \leq χ_p(G) \leq AT(G)$, where $χ(G)$ is the chromatic number, $χ_\ell(G)$ is the list chromatic number, and $χ_p(G)$ is the paint number of $G$. In this paper we…
It is known that the inequality $$ \frac{χ(G)(χ(G)-1)}{2} + |V| - χ(G) \leq |E|$$ holds for all connected graphs, where $χ(G)$ denotes the chromatic number of $G$. We prove that equality holds whenever the graph consists of a complete graph or an odd cycle, together with finitely many trees attached to its vertices.
The List-distinguishing chromatic number of graphs containing only small complete bigraphs
本文受Catlin组合技术启发，为仅含小完全二部图导出子图的连通图类建立了可区别色数与列表可区别色数的改进上界，并应用Menger定理展示了结果在基于Paley图、Dihedral群Cayley图等构造的图中的应用。
In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let $\mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let $\mathcal{C}_\text{odd}(G)$ be the set of odd numbers in $\mathcal{C}(G)$. We prove that, if $G$ has chromatic number $k$, then $\sum_{\ell\in \mathcal{C}_\text{odd}(G)}1/\ell\geq (1/2-o_k(1))\log k$. This solves Erdős and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erdős asked whether there is some $d$ such that each …
We give an upper bound for the online chromatic number of graphs with high girth and for graphs with high oddgirth generalizing Kier- stead's algorithm for graphs that contain neither a C3 or C5 as an induced subgraph.
Kostochka and Yancey proved that every $4$-critical graph $G$ has $e(G) \geq \frac{5v(G) - 2}{3}$, and that equality holds if and only if $G$ is $4$-Ore. We show that a question of Postle and Smith-Roberge implies that every $4$-critical graph with no $(7,2)$-circular-colouring has $e(G) \geq \frac{27v(G) -20}{15}$. We prove that every $4$-critical graph with no $(7,2)$-colouring has $e(G) \geq \frac{17v(G)}{10}$ unless $G$ is isomorphic to $K_{4}$ or the wheel on six vertices. We also show that if the Gallai Tree of a $4$-critical graph with no $(7,2)$-colouring has every component isomorphic…
We show that the discrete versions of the systolic inequality that estimate the number of vertices of a simplicial complex from below have substantial applications to graphs, the one-dimensional simplicial complexes. Almost directly they provide good estimates for the number of vertices of a graph in terms of its chromatic number and the length of the smallest odd cycle. Combined with the graph-theoretic techniques of Berlov and Bogdanov, the systolic approach produces even better estimates.
Strong odd coloring in minor-closed classes
本文研究小闭图类中的强奇着色问题，证明其强奇色数存在常数上界，并对外平面图几乎确定了最小常数。
Odd clique minors and chromatic bounds of {3$K_1$, paraglider}-free graphs
本文研究独立数为 2 的图的奇团 minor 猜想，并针对不含 {3$K_1$, paraglider} 的图给出改进的最优线性色数界。
We prove that for every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph with tree-width at most $k$ and odd-girth at least $g$ has circular chromatic number at most $2+\varepsilon$.
The local chromatic number of a graph was introduced by Erdős et al. [4]. In [17] a connection to topological properties of (a box complex of) the graph was established and in [18] it was shown that if a graph is strongly topologically 4-chromatic then its local chromatic number is at least four. As a consequence one obtains a generalization of the following theorem of Youngs: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs'…
Locally bipartite subgraphs via multicolor Ramsey numbers
该研究针对Erdős-Hajnal猜想，改进了$f_4(k)$的上界，还证明了奇数$g\ge5$时对应函数$h_g(k)$的增长性，证实了Mohar和Wu2018年的相关猜想。
The induced odd cycle packing number $iocp(G)$ of a graph $G$ is the maximum integer $k$ such that $G$ contains an induced subgraph consisting of $k$ pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al.~\cite{indoc} proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers $k\ge 0$ and $t\ge 1$ and any $n$-vertex g…
Odd and even cycle lengths, minimum degree and chromatic number in graphs
研究给定图中奇偶数圈长度情况下团数与色数的关系，通过证明和构造图得出相关结论，如给定奇数圈长度条件时的色数界限及特殊图类，同理对偶数圈长度也有研究，综合结果可推导多种情况下团数与色数的关系。
A class of graphs is $χ$-bounded if there is a function $f$ such that $χ(G)\le f(ω(G))$ for every induced subgraph $G$ of every graph in the class, where $χ,ω$ denote the chromatic number and clique number of $G$ respectively. In 1987, Gyárfás conjectured that for every $c$, if $\mathcal{C}$ is a class of graphs such that $χ(G)\le ω(G)+c$ for every induced subgraph $G$ of every graph in the class, then the class of complements of members of $\mathcal{C}$ is $χ$-bounded. We prove this conjecture. Indeed, more generally, a class of graphs is $χ$-bounded if it has the property that no graph in th…
On strong odd colorings of graphs
本文研究图的强奇着色问题，回答了平面图强奇色数是否存在常数界这一公开问题，并进一步探讨了树、单圈图和图乘积上的强奇着色性质。
The {\em chromatic edge-stability number} $es_χ(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph with the chromatic number smaller than that of $G$. A graph $G$ is called {\em $(3,2)$-critical} if $χ(G)=3$, $es_χ(G)=2$ and for any edge $e\in E(G)$, $es_χ(G-e)<es_χ(G)$. In this paper, we characterize $(3,2)$-critical graphs which contain at least five odd cycles. This answers a question proposed by Brešar, Klavžar and Movarraei in [Critical graphs for the chromatic edge-stability number, {\it Discrete Math.} {\bf 343}(2020) 111845].
Disproof of the Odd Hadwiger Conjecture
该研究通过证明存在特定图不包含奇极小且色数较高，从而否定了奇Hadwiger猜想。
Kierstead, Szemerédi, and Trotter showed that a graph with at most $\lfloor r/(2n)\rfloor^n$ vertices such that each ball of radius $r$ in it is $c$-colorable should have chromatic number at most $n(c-1)+1$. We show that this estimate is sharp in $r$. Namely, for every $n$, $r$, and $c$ we construct a graph $G$ containing $O((2rc)^{n-1}c)$ vertices such that $χ(G)\geq n(c-1)+1$, although each ball of radius $r$ in $G$ is $c$-colorable. The core idea is the construction of a graph whose neighborhood complex is homotopy equivalent to the join of neighborhood complexes of two given graphs.
Totally odd subdivisions in Kneser graphs
本文加强了 Kneser 图中关于 Odd Hadwiger Conjecture 的已有结果，证明了存在包含任意大完全全奇分拆的 $t$-色 Kneser 图，且所有 Kneser 图均含 $K_{χ(G)}$ 的全奇分拆，并扩展至全奇浸入研究。
A long-standing conjecture asserts that there exists a constant $c>0$ such that every graph of order $n$ without isolated vertices contains an induced subgraph of order at least $cn$ with all degrees odd. Scott (1992) proved that every graph $G$ has an induced subgraph of order at least $|V(G)|/(2χ(G))$ with all degrees odd, where $χ(G)$ is the chromatic number of $G$, this implies the conjecture for graphs with { bounded} chromatic number. But the factor $1/(2χ(G))$ seems to be not best possible, for example, Radcliffe and Scott (1995) proved $c=\frac 23$ for trees, Berman, Wang and Wargo (19…
An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree $2$ and two vertices have in-degree $2$).
Hadwiger's conjecture states that every $K_t$-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 $K_t$-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 $K_t$-minor is $f(t)$-colorable, then every graph without an odd $K_t$-minor is $2f(t)$-colorable. Using this, th…
We study graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. We establish some basic properties of the independence set polytope in the graphon setting, and duality properties between the fractional chromatic number and the fractional clique number. We present a notion of perfect graphons and characterize them in terms of induced densities of odd cycles and its complements.
On local antimagic chromatic numbers of the join of two special families of graphs
本文通过构造$(2m+1) \times (2k+1)$矩阵，完全确定了零图与奇数分支1-正则图的联图的局部反魔幻色数，并由此得到无穷多个局部反魔幻色数为3的三部图。
Let $ϕ(k)$ 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 $ϕ(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is known that $ϕ(k) \le 2k+3$ and it is conjectured by Noel that $ϕ(k) = 2k+3$. For a multi-set $λ=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a $k_λ$-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 $…
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