Related papers: Certifying coloring algorithms for graphs without …

For a fixed graph $H$, what is the smallest number of colours $C$ such that there is a proper edge-colouring of the complete graph $K_n$ with $C$ colours containing no two vertex-disjoint colour-isomorphic copies, or repeats, of $H$? We…

In 2012, L\'ev\^eque, Maffray, and Trotignon conjectured that each graph $G$ that contains no induced subdivision of $K_4$ is $4$-colorable. In this paper, we prove that this conjecture holds when $G$ contains a $K_{1,2,3}$.

A graph coloring has bounded clustering if each monochromatic component has bounded size. Equivalently, it is a partition of the vertices into induced subgraphs with bounded size components. This paper studies clustered colorings of graphs,…

An edge-coloring of a graph $G$ with colors $1,...,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In 1991 Erd\H{o}s…

In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e.,…

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively $k$-choosable if given lists of at least $k$ colours at each…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors…

Let $\partial_H(u)$ be the set of edges incident with a vertex $u$ in the graph $H$. We say that a graph $G$ is $H$-colorable if there exist total functions $f : E(G) \rightarrow E(H)$ and $g : V(G) \rightarrow V(H)$ such that $f$ is a…

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring…

Erd\H{o}s and Simonovits asked the following question: For an integer $r\geq 2$ and a family of non-bipartite graphs $\mathcal{H}$, determine the infimum of $\alpha$ such that any $\mathcal{H}$-free $n$-vertex graph with minimum degree at…

For a fixed graph $H$, in the graph homomorphism problem, denoted by $Hom(H)$, we are given a graph $G$ and we have to determine whether there exists an edge-preserving mapping $\varphi: V(G) \to V(H)$. Note that $Hom(C_3)$, where $C_3$ is…

A proper coloring of a graph is \emph{proper conflict-free} if every non-isolated vertex $v$ has a neighbor whose color is unique in the neighborhood of $v$. A proper coloring of a graph is \emph{odd} if for every non-isolated vertex $v$,…

A vertex colouring of a graph is \emph{nonrepetitive on paths} if there is no path $v_1,v_2,...,v_{2t}$ such that v_i and v_{t+i} receive the same colour for all i=1,2,...,t. We determine the maximum density of a graph that admits a…

As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every…

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We…

We show that, for every fixed positive integers $r$ and $k$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} admits a polynomial-time algorithm on $kP_3$-free graphs. This problem is a common generalization of \textsc{Max-Weight…

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive…

In their 1997 paper titled ``Fruit Salad", Gy\'{a}rf\'{a}s posed the following conjecture: there exists a constant $k$ such that if each path of a graph spans a $3$-colourable subgraph, then the graph is $k$-colourable. It is noted that…

In this paper, we prove that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number. It is well-known that even-hole-free graphs are $\chi$-bounded but we allow here the existence of…

Paths $P_1,\ldots,P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if…