Related papers: Long properly coloured cycles in edge-coloured gra…

Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…

A graph $G$ is called a complete $k$-partite ($k\geq 2$) graph if its vertices can be partitioned into $k$ independent sets $V_{1},...,V_{k}$ such that each vertex in $V_{i}$ is adjacent to all the other vertices in $V_{j}$ for $1\leq…

The least $k$ admitting a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ of a graph $G=(V,E)$ without isolated edges such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every $uv\in E$ is denoted by $\chi'_{\Sigma}(G)$. It has been…

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph is colored. A path in a total-colored graph is a {\it total proper path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any two…

A graph $G$ is $(1,3)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ so that every vertex in $G[V_1]$ has degree at most $1$ and every vertex in $G[V_2]$ has degree at most $3$. We prove that every graph with…

In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that…

We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are…

We consider a robust variant of Dirac-type problems in $k$-uniform hypergraphs. For instance, we prove that if $H$ is a $k$-uniform hypergraph with minimum codegree at least $(1/2 + \gamma )n$, $\gamma >0$, and $n$ is sufficiently large,…

A rainbow matching in an edge-coloured graph is a matching such that its edges have distinct colours. We show that every properly edge-coloured graph $G$ with $|G| \ge (9\delta(G) -5)/2$ has a rainbow matching of size $\delta(G)$, improving…

For an edge-colored complete graph, we define the color degree of a node as the number of colors appearing on edges incident to it. In this paper, we consider colorings that don't contain tricolored triangles (also called rainbow…

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $\Delta$, the…

We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned into $k$ subgraphs of maximum degree at most $d$ and denote as $\chi'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. We prove that for every…

A famous conjecture of Caccetta and H\"{a}ggkvist (CHC) states that a directed graph $D$ with $n$ vertices and minimum outdegree at least $r$ has a directed cycle of length at most $\lceil \frac{n}{r}\rceil$. In 2017, Aharoni proposed the…

A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a…

For a connected graph, we define the proper-walk connection number as the minimum number of colors needed to color the edges of a graph so that there is a walk between every pair of vertices without two consecutive edges having the same…

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb_p(G) of a graph G as the…

Let $G = (V,E)$ be an $n$-vertex graph and let $c: E \to \mathbb{N}$ be a coloring of its edges. Let $d^c(v)$ be the number of distinct colors on the edges at $v \in V$ and let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$. H. Li proved…

A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper…

An edge colouring $c$ of a graph $G$ is called conflic-free if every non-isolated edge of $G$ has a uniquely coloured neighbour in its open edge neighbourhood. The least number of colours admitting such a colouring is denoted by $\chi'_{\rm…

A graceful $l$-coloring of a graph $G$ is a proper vertex coloring with $l$ colors which induces a proper edge coloring with at most $l-1$ colors, where the color for an edge $ab$ is the absolute difference between the colors assigned to…