Related papers: Improved Upper Bounds on $a'(G\Box H)$

We show that if $G$ is a $d$-regular Vizing-class-1 graph, then the proper additive chromatic index of $G$, denoted $\eta'_p(G)$, is equal to its chromatic index. This verifies that a strengthening of the Additive Coloring Conjecture of…

Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the…

Lehel conjectured that in every $2$-coloring of the edges of $K_n$, there is a vertex disjoint red and blue cycle which span $V(K_n)$. \L uczak, R\"odl, and Szemer\'edi proved Lehel's conjecture for large $n$, Allen gave a different proof…

The cycle double cover conjecture is a long standing problem in graph theory, which links local properties, the valency of a vertex and no bridges, and a global property of the graph, being covered by a particular set of cycles. We prove…

Given a graphic degree sequence $D$, let $\chi(D)$ (respectively $\omega(D)$, $h(D)$, and $H(D)$) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique…

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…

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let $\mathcal{A}(G)$ be…

Let $G$ be a graph with maximum degree $\Delta$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours…

Given a graph $G$ possibly with multiple edges but no loops, denote by $\Delta$ the {\it maximum degree}, $\mu$ the {\it multiplicity}, $\chi'$ the {\it chromatic index} and $\chi_f'$ the {\it fractional chromatic index} of $G$,…

Let $K_n^c$ be an edge-coloured complete graph on $n$ vertices. Let $\Delta_{\rm mon}(K_n^c)$ denote the largest number of edges of the same colour incident with a vertex of $K_n^c$. A properly coloured cycle is a cycle such that no two…

For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $\chi'_{int}(G,X)$ the…

A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges…

Let $\chi_l(G)$ denote the list chromatic number of the $r$-uniform hypergraph~$G$. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if $G$ is simple and $d$-regular, then…

A classical Tur\'an problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph $H$ as a subgraph. It is well-known that the chromatic number of $H$ is the graph parameter which…

Given a graph $G$, denote by $\Delta$, $\bar{d}$ and $\chi^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$…

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes…

The Linear Arboricity Conjecture asserts that the linear arboricity of a graph with maximum degree $\Delta$ is $\lceil (\Delta+1)/2 \rceil$. For a $2k$-regular graph $G$, this implies $la(G) = k+1$. In this note, we utilize a network flow…

Casselgren, Markst\"orm, and Pham conjectured that any precolored dis\-tan\-ce-2 matching in the $d$-dimensional cube $Q_d$ with at most $d$ colors can be extended to a proper $d$-edge-coloring. In this paper, we prove this conjecture and…

Given a graph $G$, let $\Delta_2(G)$ denote the maximum number of neighbors any two distinct vertices of $G$ have in common. Vu (2002) proposed that, provided $\Delta_2(G)$ is not too small as a proportion of the maximum degree $\Delta(G)$…

The page number of a directed acyclic graph $G$ is the minimum $k$ for which there is a topological ordering of $G$ and a $k$-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the…