Related papers: The absence of monochromatic triangle implies vari…
For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a…
Let $G$ be an edge-coloured graph. The minimum colour degree $\delta^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly…
Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is defined as the number of colors of the edges incident to $v$. The color number of $G$ is defined as the number of colors of the edges in $G$. A rainbow triangle…
In this note we consider a Ramsey property of random $d$-regular graphs, $\mathcal{G}(n,d)$. Let $r\ge 2$ be fixed. Then w.h.p. the edges of $\mathcal{G}(n, 2r)$ can be colored such that every monochromatic component has size $o(n)$. On the…
For two graphs, $G$, and $H$, an edge-coloring of a complete graph is $(G,H)$-good if there is no monochromatic subgraph isomorphic to $G$ and no rainbow subgraph isomorphic to $H$ in this coloring. The set of number of colors used by some…
An edge-colouring of a graph $G$ can fail to be rainbow for two reasons: either it contains a monochromatic cherry (a pair of incident edges), or a monochromatic matching of size two. A colouring is a proper colouring if it forbids the…
The rainbow arborescence conjecture posits that if the arcs of a directed graph with $n$ vertices are colored by $n-1$ colors such that each color class forms a spanning arborescence, then there is a spanning arborescence that contains…
Let D be a finite set of positive real numbers. The distance graph G(R,D) is the graph with vertex set R (set of real numbers), and two vertices x, y are adjacent if |x-y| belongs to D. We prove that every positive integer t>1 there is a…
An edge-colored rooted directed tree (aka arborescence) is path-monochromatic if every path in it is monochromatic. Let $k,\ell$ be positive integers. For a tournament $T$, let $f_T(k)$ be the largest integer such that every $k$-edge…
A $c$-edge-colored multigraph has each edge colored with one of the $c$ available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two…
An edge-coloured path is monochromatic if all of its edges have the same colour. For a $k$-connected graph $G$, the monochromatic $k$-connection number of $G$, denoted by $mc_k(G)$, is the maximum number of colours in an edge-colouring of…
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then.…
Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphs $G$: every…
Following problems posed by Gy\'arf\'as, we show that for every $r$-edge-colouring of $K_n$ there is a monochromatic triple star of order at least $n/(r-1)$, improving a previous result by Ruszink\'o. An edge colouring of a graph is called…
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…
A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions…
In this short note, we study pairwise edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, where a graph is rainbow if its edges have distinct colours. Brualdi and Hollingsworth conjectured that every $K_n$…
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 star coloring of a graph $G$ is a proper vertex coloring such that the subgraph induced by any pair of color classes is a star forest. The star chromatic number of $G$ is the minimum number of colors needed to star color $G$. In this…
An edge-colored graph $G$ is called rainbow if every edge of $G$ receives a different color. The anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an…