Related papers: Decomposing edge-coloured complete symmetric digra…

A conjecture due to the fourth author states that every $d$-regular planar multigraph can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and its complement. For $d = 3$ this…

We address an old (1977) conjecture of a subset of the authors (a variant of Ryser's conjecture): in every r-coloring of the edges of a biclique [A,B] (complete bipartite graph), the vertex set can be covered by the vertices of at most 2r-2…

A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every…

A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call $k$ paths $P_1,\cdots,P_k$ rainbow monochromatic paths if every $P_i$ is monochromatic and for any two $i\neq j$, $P_i$ and…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph…

Let $G$ be a simple graph that is properly edge coloured with $m$ colours and let $\M=\{M_1,\ldots, M_m\}$ be the set of $m$ matchings induced by the colours in $G$. Suppose that $m\le n-n^{c}$, where $c>9/10$, and every matching in $\M$…

Let $c$ be an edge-colouring of a graph $G$ such that for every vertex $v$ there are at least $d \ge 2$ different colours on edges incident to $v$. We prove that $G$ contains a properly coloured path of length 2d or a properly coloured…

Answering a question raised by Dudek and Pra\l{}at, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that…

Given an $r$-edge-colouring of the edges of a graph $G$, we say that it can be partitioned into $p$ monochromatic cycles when there exists a set of $p$ vertex-disjoint monochromatic cycles covering all the vertices of $G$. In the literature…

A classical result of Corr\'adi and Hajnal states that every graph $G$ on $n$ vertices with $n\in 3\mathbb{N}$ and $\delta(G) \ge 2n/3$ contains a perfect triangle-tiling, i.e.,\ a spanning set of vertex-disjoint triangles. We explore a…

Let $k$ and $r$ be two integers with $k \ge 2$ and $k\ge r \ge 1$. In this paper we show that (1) if a strongly connected digraph $D$ contains no directed cycle of length $1$ modulo $k$, then $D$ is $k$-colorable; and (2) if a digraph $D$…

A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can…

In this paper, we present a foundation study for proper colouring of edge-set graphs. The authors consider that a detailed study of the colouring of edge-set graphs corresponding to the family of paths is best suitable for such foundation…

In this paper we describe all edge-colored graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. They correspond to fully symmetric homogeneous factorizations of complete graphs. Our…

We prove #P-completeness results for counting edge colorings on simple graphs. These strengthen the corresponding results on multigraphs from [4]. We prove that for any $\kappa \ge r \ge 3$ counting $\kappa$-edge colorings on $r$-regular…

We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph…

We study a Tur\'an-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a member from…

In the Coloured Clustering problem, we wish to colour vertices of an edge coloured graph to produce as many stable edges as possible, i.e., edges with the same colour as their ends. In this paper, we reveal that the problem is in fact a…

In an $r$-coloring of edges of the complete graph on $n$ vertices, how many edges are there in the largest monochromatic connected component? A construction of Gy\'arf\'as shows that for infinitely many values of $r$, there exist colorings…

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in…