Related papers: Towards on-line Ohba's conjecture

For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of graphs which have girth $2\ell+1$ and have no odd hole with length greater than $2\ell+1$. Plummer and Zha conjectured that every 3-connected and internally…

In the $\ell$-Coloring Problem, we are given a graph on $n$ nodes, and tasked with determining if its vertices can be properly colored using $\ell$ colors. In this paper we study below-guarantee graph coloring, which tests whether an…

Let G be a simple connected graph. Then chi L(G) and chi D(G) will denote the locating chromatic number and the distinguishing chromatic number of G, respectively. In this paper, we investigate a comparison between chi L(G) and chi D(G). In…

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours,…

Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from…

We prove that for every oriented graph $D$ and every choice of positive integers $k$ and $\ell$, there exists an oriented graph $D^*$ along with a surjective homomorphism $\psi\colon V(D^*) \to V(D)$ such that: (i) girth$(D^*) \geq\ell$;…

Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $\Delta(G)\ge 9$ and its clique number satisfies $\omega(G)\le \Delta(G)-1$, then its chromatic number satisfies $\chi(G) \le \Delta(G)-1$. We prove this…

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per…

The chromatic polynomial of a graph G counts the number of proper colorings of G. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a…

In a seminal paper, Alon and Tarsi have introduced an algebraic technique for proving upper bounds on the choice number of graphs (and thus, in particular, upper bounds on their chromatic number). The upper bound on the choice number of $G$…

Let $G$ be a simple graph with maximum degree $\Delta$. We call $G$ \emph{overfull} if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$.…

A well-known theorem of Vizing states that if $G$ is a simple graph with maximum degree $\Delta$, then the chromatic index $\chi'(G)$ of $G$ is $\Delta$ or $\Delta+1$. A graph $G$ is class 1 if $\chi'(G)=\Delta$, and class 2 if…

For integers $k\ge 1$ and $m\ge 2$, let $g(k,m)$ be the least integer $n\ge 1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. We prove that \[ g(k,m)\le…

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $\chi_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge…

The distinguishing chromatic number of a graph $G$, denoted $\chi_D(G)$, is the minimum number of colours in a proper vertex colouring of $G$ that is preserved by the identity automorphism only. Collins and Trenk proved that $\chi_D(G)\le…

We prove that $K_{\chi(G)}$ is the only critical graph $G$ with $\chi(G) \geq \Delta(G) \geq 6$ and $\omega(\mathcal{H}(G)) \leq \left \lfloor \frac{\Delta(G)}{2} \right \rfloor - 2$. Here $\mathcal{H}(G)$ is the subgraph of $G$ induced on…

A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a $P_5$-free graph with clique number $\omega\ge 3$ has chromatic number at most $\omega^{\log_2(\omega)}$. The best previous result was an exponential upper…

An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $\chi_o(G)$, is the minimum number of colors that admits an…

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in $[1,\sigma]$. For $\sigma=1$ it is the class of unit interval graphs, and for $\sigma=\infty$ the class of all interval graphs.…

A famous conjecture of Gy\'arf\'as and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss…