Related papers: On graphs with no induced $P_5$ or $K_5-e$

Let $G$ be a graph. We say that $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. We use $P_t$ and $C_t$ to denote a path…

We prove that the class of $(K_t,sP_1+P_5)$-free graphs has bounded mim-width for every $s\geq 0$ and $t\geq 1$, and that there is a polynomial-time algorithm that, given a graph in the class, computes a branch decomposition of constant…

This article foucuses on $(P_3\cup P_2,K_4)$-free graph. In this paper, we prove that if G is $(P_3\cup P_2,K_4)$-free, then $\chi(G)\le 7$. We then use our result to obtain the upper bound of order and chromatic number of…

If a graph has no induced subgraph isomorphic to any graph in a finite family $\{H_1,\ldots,H_p\}$, it is said to be $(H_1,\ldots,H_p)$-free. The class of $H$-free graphs has bounded clique-width if and only if $H$ is an induced subgraph of…

Resolving a 1985 open problem of Gy\'arf\'as, we prove that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path $P_5$. Our approach introduces a chromatic density framework involving…

The $k$-Coloring problem on hereditary graph classes has been a deeply researched problem over the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs. We say that a…

A graph $G$ is $k$-vertex-critical if $\chi(G)=k$, but $\chi(G')<k$ for every proper induced subgraph $G'$ of $G$. For a family of graphs $\mathcal{F}$, $G$ is $\mathcal{F}$-free if no graph $F \in \mathcal{F}$ is an induced subgraph of…

Given a graph $G$, the strong clique number of $G$, denoted $\omega_S(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned…

We show that the following problems are NP-complete. 1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph? 2. Is the difference between the chromatic number and clique number at most $1$…

A hereditary class $\cal G$ of graphs is {\em $\chi$-bounded} if there is a {\em $\chi$-binding function}, say $f$, such that $\chi(G)\le f(\omega(G))$ for every $G\in\cal G$, where $\chi(G)(\omega(G))$ denotes the chromatic (clique) number…

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In this paper, we determine the order of magnitude of the clique chromatic number of the random graph…

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. A $P_t$ is the path on $t$ vertices. A chair is a $P_4$ with an additional vertex adjacent to one of the…

We give a polynomial-time algorithm that computes the chromatic number of any graph that contains no path on five vertices and no bull as an induced subgraph (where the bull is the graph with five vertices $a,b,c,d,e$ and edges…

Daligault, Rao and Thomass\'e asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (JCTB 2017+) gave a negative answer to this question,…

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…

In this paper, we are interested in $4$-colouring algorithms for graphs that do not contain an induced path on $6$ vertices nor an induced bull, i.e., the graph with vertex set $\{v_1,v_2,v_3,v_4,v_5\}$ and edge set…

The clique chromatic number of a graph G=(V,E) is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random…

As usual, $P_n$ ($n \geq 1$) denotes the path on $n$ vertices, and $C_n$ ($n \geq 3$) denotes the cycle on $n$ vertices. For a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is…

We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast…

The chromatic number of signed graphs is defined recently. The coloring and clique problem of interval graphs has been studied and polynomial time algorithms are established. Here we consider these problems for signed interval graphs and…