Related papers: Bull-free graphs and $\chi$-boundedness

This paper discusses the complete structure of the (bull, diamond)-free graphs. As an application of that, we give the characterization of the partitionable (bull, diamond)-free graphs. Moreover, we show that such a partition for a…

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon, Krivelevich,…

A graph is unichord free if it does not contain a cycle with exactly one chord as its subgraph. In [3], it is shown that a graph is unichord free if and only if every minimal vertex separator is a stable set. In this paper, we first show…

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $\beta(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a…

Considering connected $K_{1,3}$-free graphs with independence number at least $3$, Chudnovsky and Seymour (2010) showed that every such graph, say $G$, is $2\omega$-colourable where $\omega$ denotes the clique number of $G$. We study…

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. Brandst\"adt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandst\"adt, Le and Mosca…

A graph class is $\chi$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erd\H{o}s conjectured that intersection graphs of straight-line segments in the plane are…

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…

Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum cardinality of…

We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $\epsilon:=\epsilon(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from…

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. This is sharp up to a factor…

We study the Max Partial $k$-Coloring problem, where we are given a vertex-weighted graph, and we ask for a maximum-weight induced subgraph that admits a proper $k$-coloring. For $k=1$ this problem coincides with Maximum Weight Independent…

Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Let $S_{t,t,t}$ be the graph obtained from $K_{1,3}$ by subdividing each edge…

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…

A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph…

Goedgebeur and Schaudt [J. Graph Theory 87 (2018) 188-207] conjectured that all 4-vertex-critical $(P_7,C_3)$-free graphs belongs to the family $\cal G$, which consists of seven explicitly defined graphs. In this paper, we establish a…

For a graph $G$, let $\chi(G)$ ($\omega(G)$) denote its chromatic (clique) number. A $P_2+P_3$ is the graph obtained by taking the disjoint union of a two-vertex path $P_2$ and a three-vertex path $P_3$. A $\bar{P_2+P_3}$ is the complement…

For an integer $t$, a graph $G$ is called {\em{$C_{>t}$-free}} if $G$ does not contain any induced cycle on more than~$t$ vertices. We prove the following statement: for every pair of integers $d$ and $t$ and a CMSO$_2$ statement~$\phi$,…

A graph $G = (V, E)$ is \emph{partitionable} if there exists a partition $\{A, B\}$ of $V$ such that $A$ induces a disjoint union of cliques and $B$ induces a triangle-free graph. In this paper we investigate the computational complexity of…

Let $P_n$ and $K_n$ denote the induced path and complete graph on $n$ vertices, respectively. The {\em kite} is the graph obtained from a $P_4$ by adding a vertex and making it adjacent to all vertices in the $P_4$ except one vertex with…