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 number of graph classes. In particular, we consider for a number of classes whether they have bounded odd chromatic number. Our main results are that interval graphs, graphs of bounded modular-width and graphs of bounded maximum degree all have bounded odd chromatic number.
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…
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…
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…
A signed graph $(G, σ)$ is a graph $G$ along with a function $σ: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph $(G, σ)$ is the minimum number of vertices $|V(H)|$ of a signed graph $(H, π)$ to which $(G, σ)$ admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention …
A hereditary class $\mathcal{G}$ of graphs is $χ$-bounded if there is a $χ$-binding function, say $f$ such that $χ(G) \leq f(ω(G))$, for every $G \in \cal{G}$, where $χ(G)$ ($ω(G)$) denote the chromatic (clique) number of $G$. It is known that for every $2K_2$-free graph $G$, $χ(G) \leq \binom{ω(G)+1}{2}$, and the class of ($2K_2, 3K_1$)-free graphs does not admit a linear $χ$-binding function. In this paper, we are interested in classes of $2K_2$-free graphs that admit a linear $χ$-binding function. We show that the class of ($2K_2, H$)-free graphs, where $H\in \{K_1+P_4, K_1+C_4, \overline{P…
A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper …
Robin Thomas asked whether for every proper minor-closed class C, there exists a polynomial-time algorithm approximating the chromatic number of graphs from C up to a constant additive error independent on the class C. We show this is not the case: unless P=NP, for every integer k>=1, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k>=1 and 1<=β<=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta.chi(G)+(4-3beta)k colors. As far as we know, this is the fir…
The packing chromatic number $χ$ $ρ$ (G) of a graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V 1 ,. .. , V k , in such a way that every two distinct vertices in V i are at distance greater than i in G for every i, 1 $\le$ i $\le$ k. Recently, Balogh, Kostochka and Liu proved that $χ$ $ρ$ is not bounded in the class of subcubic graphs [Packing chromatic number of subcubic graphs, Discrete Math. 341 (2018), 474483], thus answering a question previously addressed in several papers. However, several subclasses of cubic or subcubic g…
In 1998, Reed conjectured that for any graph $G$, $χ(G) \leq \lceil \frac{ω(G) + Δ(G)+1}{2}\rceil$, where $χ(G)$, $ω(G)$, and $Δ(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$. In this paper, we study this conjecture for some expansions of graphs, that is graphs obtained with the well known operation composition of graphs. We prove that Reed's Conjecture holds for expansions of bipartite graphs, for expansions of odd holes where the minimum chromatic number of the components is even, when some component of the expansion has chromatic number 1 or w…
In this paper, we have obtained the total chromatic as well as equitable and neighborhood sum distinguishing total chromatic numbers of some classes of the circulant graphs.
Although it has recently been proved that the packing chromatic number is unbounded on the class of subcubic graphs, there exists subclasses in which the packing chromatic number is finite (and small). These subclasses include subcubic trees, base-3 Sierpi{ń}ski graphs and hexagonal lattices.In this paper we are interested in the packing chromatic number of subcubic outerplanar graphs. We provide asymptotic bounds depending on structural properties of the outerplanar graphs and determine sharper bounds for some classes of subcubic outerplanar graphs.
A {\em snark} is a cubic cyclically 4-edge connected graph with edge chromatic number four and girth at least five. We say that a graph $G$ is {\em odd 2-factored} if for each 2-factor F of G each cycle of F is odd. In this paper, we present a method for constructing odd 2--factored snarks. In particular, we construct two families of odd 2-factored snarks that disprove a conjecture by some of the authors. Moreover, we approach the problem of characterizing odd 2-factored snarks furnishing a partial characterization of cyclically 4-edge connected odd 2-factored snarks. Finally, we pose a new co…
Balanced-chromatic number and Hadwiger-like conjectures
Universidad de Valparaíso（瓦尔帕莱索大学）；Millennium Nucleus for Social Data Science (SODAS)（社会数据科学千年核（SODAS））；Auburn University（奥本大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息研究与研究所）；École Normale Supérieure, Paris（巴黎高等师范学院）
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that vertices of $G$ can be partitioned into disjoint classes $X_1, ..., X_k$ where vertices in $X_i$ have pairwise distance greater than $i$. We study the packing chromatic number of infinite distance graphs $G(Z, D)$, i.e. graphs with the set $Z$ of integers as vertex set and in which two distinct vertices $i, j \in Z$ are adjacent if and only if $|i - j| \in D$. In this paper we focus on distance graphs with $D = \{1, t\}$. We improve some results of Togni who initiated the study. It is shown that $χ_ρ(G(Z…
The b-chromatic number of a graph G is the largest integer k such that G has a coloring of the vertices in k color classes such that every color class contains a vertex that has a neighbour in all other color classes. We characterize the class of chordal graphs for which the b-chromatic number is equal to the chromatic number for every induced subgraph.
An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Goldberg conjecture for multigraphs. The results obtained in the paper support our con…
Erdös conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound (\cite{gyarfas}, Problem 5.1.55 in \cite{West}) was so far known. We make progress on this conjecture by showing that $G$ contains an odd cycle of length at least $O(k\log\log k)$. Erdös' conjecture is known to hold for graphs with girth at least 5. We show that if a girth 4 graph is $C_5$ free, then Erdös' conjecture holds. When the number of vertices is not too larg…
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set $V(G)$ can be partitioned into disjoint classes $X_1, ..., X_k$, where vertices in $X_i$ have pairwise distance greater than $i$. For the 2-dimensional square lattice $\mathbb{Z}^2$ it is proved that $χ_ρ(\mathbb{Z}^2) \geq 12$, which improves the previously known lower bound 10.
Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most $3$. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of arrangements of (pseudo-)circles. In this paper the conjecture is verified for $\triangle$-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which ma…
A normal odd partition T of the edges of a cubic graph is a partition into trails of odd length (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition and internal in some trail. For each vertex v, we can distinguish the edge for which this vertex is pending. Three normal odd partitions are compatible whenever these distinguished edges are distinct for each vertex. We examine this notion and show that a cubic 3 edge-colorable graph can always be provided with three compatible normal odd partitions. The Petersen graph has this property and we can constr…
