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 a question of Moore and West.
Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$. We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for $5$-chromatic graphs as follows: Every $5$-chromatic graph contains a $K_5$-mi…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos (Combinatorica, 2006), Simonyi, Tardif, and Zsbán (The Electronic Journal of Combinatorics, 2013), and Chen (Journal of Combinatorial Theory, Series A, 2011). As a consequence of the generalization of Chen's theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisf…
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.
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.
Every properly colored graph with $χ(G)=k$ colors has edge-disjoint Kempe "backbones", Kempe chains anchored by color-critical vertices for each pair of colors. Certain color permutations arrange these backbones into a clique-like structure, a strengthening of strong immersions of complete graphs. This strengthened immersion is suggested as a template for identifying the disjoint subgraphs comprising Hadwiger's conjectured $K_k$ minor present in $k$-chromatic graphs.
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 …
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.
The chromatic critical edge theorem of Simonovits states that for a given color critical graph $H$ with $χ(H)=k+1$, there exists an $n_0(H)$ such that the Turán graph $T_{n,k}$ is the only extremal graph with respect to $ex(n,H)$ provided $n \geq n_0(H)$. Nikiforov's pioneer work on spectral graph theory implies that the color critical edge theorem also holds if $ex(n,H)$ is replaced by the maximum spectral radius and $n_0(H)$ is an exponential function of $|H|$. We want to know which color critical graphs $H$ satisfy that $n_0(H)$ is a linear function of $|H|$. Previous graphs include complet…
It is well known that a random subgraph of the complete graph $K_n$ has chromatic number $Θ(n/\log n)$ w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any $n$-chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least $n$, the fractional chromatic number of its random subgraph is at least $n/(8\log_2(4n))$ with probability more than $1-\frac{1}{2n}$. This gives the affirmative answer for a strengthening of Bukh's question for the fractional chromatic number.
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 $β> 1/4$.
Let H be a tree. It was proved by Rodl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$ as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Pilipczuk, Rzazewski, Thomasse and Walczak.
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 $7$.
In 1971, Tomescu conjectured [Le nombre des graphes connexes $k$-chromatiques minimaux aux sommets étiquetés, C. R. Acad. Sci. Paris 273 (1971), 1124--1126] that every connected graph $G$ on $n$ vertices with $χ(G) = k \geq 4$ has at most $k!(k-1)^{n-k}$ $k$-colourings, where equality holds if and only if the graph is formed from $K_k$ by repeatedly adding leaves. In this note we prove (a strengthening of) the conjecture of Tomescu when $k=5$.
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…
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…
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…
It is proved that every connected graph $G$ on $n$ vertices with $χ(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed from $K_4$ by repeatedly adding leaves. This confirms (a strengthening of) the $4$-chromatic case of a long-standing conjecture of Tomescu [Le nombre des graphes connexes $k$-chromatiques minimaux aux sommets etiquetes, C. R. Acad. Sci. Paris 273 (1971), 1124-1126]. Proof methods may be of independent interest. In particular, one of our auxiliary results about …
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.
A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges receive different colours and every vertex and incident edge receive different colours. This paper considers a strengthening of this condition and examines the minimum number of colours required for a total colouring with the additional property that for any adjacent vertices $u$ and $v$, the sets of colours incident to $u$ is different from the set of colours i…
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…
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…
Kneser's 1955 conjecture -- proven by Lovász in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser's conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász' result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser's conjecture. Here we prove a common generalization of a…
On the Turán number of odd-ballooning of $3$-chromatic graphs
School of Mathematics, East China University of Science and Technology（华东理工大学数学学院）；School of Mathematics and Finance, Chuzhou University（滁州学院数学与金融学院）；School of Finance Business, Shanghai Normal University（上海师范大学商学院）
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…
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…
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.
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.
The Gyárfás-Sumner conjecture says that for every forest $H$ and every integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices then it has bounded chromatic number. (A graph is $H$-free if it does not contain an induced copy of $H$.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest $H$, if $G$ is $H$-free and does not contain $K_{t,t}$ as a subgraph then it has bounded…
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…
Let $n,k,r$ be positive integers with $n \geq rk$ and $r \geq 2$. Consider a circle $C$ with~$n$ points~$1,\ldots,n$ in clockwise order. The $r$-stable \emph{interlacing graph} $\text{IG}_{n,k}^{(r)}$ is the graph with vertices corresponding to $k$-subsets $S$ of $\{1,...,n\}$ such that any two distinct points in~$S$ have distance at least~$r$ around the circle, and edges between~$k$-subsets $P$ and $Q$ if they \emph{interlace}: after removing the points in~$P$ from $C$, the points in~$Q$ are in different connected components. In this paper we prove that the circular chromatic number of $\text…
We prove that the chromatic polynomial $P_\mathbb{G}(q)$ of a finite graph $\mathbb{G}$ of maximal degree $\D$ is free of zeros for $\card q\ge C^*(\D)$ with $$ C^*(\D) = \min_{0<x<2^{1\over \D}-1} {(1+x)^{\D-1}\over x [2-(1+x)^\D]} $$ This improves results by Sokal (2001) and Borgs (2005). Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.
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.
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'…
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…
