Related papers: Partitioning edge-coloured infinite complete bipar…
Kang and Park recently showed that every cubic (loopless) multigraph is incidence 6-choosable [On incidence choosability of cubic graphs. \emph{arXiv}, April 2018]. Equivalently, every bipartite graph obtained by subdividing once every edge…
Let $G$ be a graph whose edges are coloured with $k$ colours, and $\mathcal H=(H_1,\dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $\mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either…
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 solve a recent question of Caro, Patk\'os and Tuza by determining the exact maximum number of edges in a bipartite connected graph as a function of the longest path it contains as a subgraph and of the number of vertices in each side of…
The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was…
In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado formulated a similar conjecture that an…
A graph $G$ is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=\chi(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|\leq 2\chi(G)+1$ is chromatic-choosable. Since each…
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…
We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-coloured random graph. More precisely, we show that for $p\gg n^{-1/6}{(\ln n)}^{1/6}$, in any $3$-edge-colouring of the…
P. Kirchberger proved that, for a finite subset $X$ of $\mathbb{R}^{d}$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can…
An induced matching in a graph $G$ is a matching such that its end vertices also induce a matching. A $(1^{\ell}, 2^k)$-packing edge-coloring of a graph $G$ is a partition of its edge set into disjoint unions of $\ell$ matchings and $k$…
It is well-known that an undirected graph has no odd cycle if and only if it is bipartite. A less obvious, but similar result holds for directed graphs: a strongly connected digraph has no odd cycle if and only if it is bipartite. Can this…
For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then…
A path in a vertex-colored graph is called a \emph{vertex-monochromatic path} if its internal vertices have the same color. A vertex-coloring of a graph is a \emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for short),…
In this paper, we consider a number of results and seven conjectures on properly edge-coloured (PC) paths and cycles in edge-coloured multigraphs. We overview some known results and prove new ones. In particular, we consider a family of…
A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…
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…
A strong edge coloring of a graph $G$ is a proper edge coloring in which each color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey proposed a conjecture that every bipartite graph without $4$-cycles and with the…
A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one…
A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced subgraph…