Related papers: Complements of nearly perfect graphs

Reidl, S\'anchez Villaamil, and Stravopoulos (2019) characterized graph classes of bounded expansion as follows: A class $\mathcal{C}$ closed under subgraphs has bounded expansion if and only if there exists a function $f:\mathbb{N} \to…

Let $G$ be a simple graph. Denote by $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ \emph{overfull} if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and {\it critical}…

For a class $\mathcal{G}$ of graphs, the objective of \textsc{Subgraph Complementation to} $\mathcal{G}$ is to find whether there exists a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph…

Let $\mathscr C$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known {\L}o\'s-Tarski Theorem from classical model theory implies that $\mathscr C$ is definable in first-order logic (FO) by a…

We prove a conjecture of Geelen that every proper vertex-minor-closed class of graphs is $\chi$-bounded.

Inspired by a famous characterization of perfect graphs due to Lov\'{a}sz, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $\alpha(H) + \omega(H) \geq |V(H)|$. (Here $\alpha$ and $\omega$ denote the…

For a graph $G$, $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $\chi$-bounded if there is a function $f$ such that for each graph $G$ in…

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function $f\colon\mathbb{N}\to\mathbb{N}\cup\{\infty\}$ with $f(1)=1$ and $f(n)\geq\binom{3n+1}{3}$, we construct a hereditary class of graphs…

In this paper, we characterize the class of {\em contraction perfect} graphs which are the graphs that remain perfect after the contraction of any edge set. We prove that a graph is contraction perfect if and only if it is perfect and the…

A class of graphs $\mathcal{C}$ is closed under powers if for every graph $G\in\mathcal{C}$ and every $k\in\mathbb{N}$, $G^k\in\mathcal{C}$. Also $\mathcal{C}$ is strongly closed under powers if for every $k\in\mathbb{N}$, if…

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is \emph{$\Delta$-critical} if $\chi'(G)=\Delta(G)+1$ and…

Let $G$ be a simple graph, and let $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and critical if $\chi'(H) <…

In this paper, we establish an optimal $\chi$-binding function for $(P_2\cup P_4,\text{ diamond})$-free graphs. We prove that for any graph $G$ in this class, $\chi(G)\le 4$ when $\omega(G)=2$, $\chi(G)\le 6$ when $\omega(G)=3$, and…

$\chi$-bounded classes are studied here in the context of star colorings and more generally $\chi_p$-colorings. This leads to natural extensions of the notion of bounded expansion class and to structural characterization of these. In this…

The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma…

Let $G$ be a graph and $f:V(G)\rightarrow \mathbb{N}$ be a function. An $f$-coloring of a graph $G$ is an edge coloring such that each color appears at each vertex $v\in V(G)$ at most $f (v)$ times. The minimum number of colors needed to…

Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in…

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $\rho(G)$ of $G$ is the maximum size of a vertex set whose closed…

A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $\omega(H[X_1]) < \omega(H)$ and $H[X_2]$ is a perfect graph. In this…

This work introduces the concept of \emph{upper-critical graphs}, in a complementary way of the conventional (lower)critical graphs: an element $x$ of a graph $G$ is called \emph{critical} if $\chi(G-x)<\chi(G)$. It is said that $G$ is a…