Related papers: The optimal chromatic bound for even-hole-free gra…

A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. It is NP-hard to color the vertices of an odd hole-free graph. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ with at least…

A class of graphs is $\chi$-bounded if there is a function $f$ such that $\chi(G)\le f(\omega(G))$ for every induced subgraph $G$ of every graph in the class, where $\chi,\omega$ denote the chromatic number and clique number of $G$…

A hole in a graph $G$ is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph $K_4$ by removing an edge. A pyramid is a graph consisting of a triangle…

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, $K_4$)-free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019].…

A total $k$-coloring of a graph is an assignment of $k$ colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph $G$ has a…

The HVN is a graph formed by removing two edges incident to the same vertex from the complete graph $K_5$. In this paper, we prove that every ($P_2\cup P_4$, HVN)-free graph $G$ satisfies $\chi(G)\leq\lceil\frac{4}{3}\omega(G)\rceil$ when…

An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$,…

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_s$ be the path on $t$ vertices and the cycle on $s$ vertices, respectively. In this paper we show…

For a graph $G$, $\chi(G)$ will denote its chromatic number, and $\omega(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such…

Let $\chi'_\subset(G)$ be the least number of colours necessary to properly colour the edges of a graph $G$ with minimum degree $\delta\geq 2$ so that the set of colours incident with any vertex is not contained in a set of colours incident…

It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via…

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy…

Given a graph or multigraph $G$, let $\chi'_{trans}(G)$ denote the minimum integer $n$ such that any proper $\chi'(G)$--edge coloring of $G$ can be transformed into any other proper $\chi'(G)$--edge coloring of $G$ by a series of…

Let $G$ be a graph with maximum degree $\Delta$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours…

Even-hole-free graphs are a graph class of much interest. Foley et al. [Graphs Comb. 36(1): 125-138 (2020)] have recently studied $(4K_1, C_4, C_6)$-free graphs, which form a subclass of even-hole-free graphs. Specifically, Foley et al.…

We present a class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width. In general, even-hole-free graphs have unbounded rank-width, because chordal graphs are even-hole-free. A.A. da Silva, A. Silva and…

Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly…

A bull is a graph obtained from a four-vertex path by adding a vertex adjacent to the two middle vertices of the path. A graph $G$ is bull-free if no induced subgraph of $G$ is a bull. We prove that for all $k,t\in \mathbb{N}$, if $G$ is a…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of…

Let $G$ be a claw-free graph on $n$ vertices with clique number $\omega$, and consider the chromatic number $\chi(G^2)$ of the square $G^2$ of $G$. Writing $\chi'_s(d)$ for the supremum of $\chi(L^2)$ over the line graphs $L$ of simple…