Related papers: Proper connection number of random graphs
We prove that a random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, is typically such that for any $r$-colouring of its edges there exists a Hamilton cycle with at least $(2/(r+ 1)-o(1))n$ edges of the same colour. This…
For a fixed graph $H$, what is the smallest number of colours $C$ such that there is a proper edge-colouring of the complete graph $K_n$ with $C$ colours containing no two vertex-disjoint colour-isomorphic copies, or repeats, of $H$? We…
Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph $G$ is rainbow disconnected if for every two vertices $u$ and $v$, there…
The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are…
An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an interval $t$-coloring if for each $i\in \{1,2,\ldots,t\}$ there is at least one edge of $G$ colored by $i$, and the colors of edges incident to any vertex of $G$ are…
In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper $r$-coloring $\varphi$ of a graph $G$. We investigate the problem of finding a proper $r$-coloring of $G$, which is "the…
In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G…
Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields…
A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex $x$ is called…
A vertex-colored graph $G$ is rainbow vertex-connected if any pair of distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of $G$, denoted by $rvc(G)$, is the minimum…
We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distant at most $r$ in a graph are called $r$-adjacent. The least number of colours…
We say that a graph $G$ is Ramsey for $H_1$ versus $H_2$, and write $G \to (H_1,H_2)$, if every red-blue colouring of the edges of $G$ contains either a red copy of $H_1$ or a blue copy of $H_2$. In this paper we study the threshold for the…
The square $G^2$ of a graph $G$ is the graph on $V(G)$ with a pair of vertices $uv$ an edge whenever $u$ and $v$ have distance $1$ or $2$ in $G$. Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever the…
A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A path in a total-colored graph is a {\it total monochromatic path} if all the edges and internal vertices on the path have the same…
An oriented graph is a directed graph with no bi-directed edges, i.e. if $xy$ is an edge then $yx$ is not an edge. The oriented size Ramsey number of an oriented graph $H$, denoted by $r(H)$, is the minimum $m$ for which there exists an…
For a given $\delta \in (0,1)$, the randomly perturbed graph model is defined as the union of any $n$-vertex graph $G_0$ with minimum degree $\delta n$ and the binomial random graph $\mathbf{G}(n,p)$ on the same vertex set. Moreover, we say…
We consider the following random model for edge-colored graphs. A graph $G$ on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting each edge of $G$ remain independently with probability $p$. Then, each edge of $G_p$ is…
An {\em odd subgraph} of a graph is a subgraph in which every vertex has odd degree. A graph $G$ is said to be {\em odd $k$-edge-colorable} if there exists an edge-coloring $E(G) \rightarrow \{1,2, \ldots, k\}$ such that each non-empty…
A path in an edge-colored graph is \textit{rainbow} if no two edges of it are colored the same. The graph is said to be \textit{rainbow connected} if there is a rainbow path between every pair of vertices. If there is a rainbow shortest…
The Ramsey number $r(G)$ of a graph $G$ is the minimum number $N$ such that any red-blue colouring of the edges of $K_N$ contains a monochromatic copy of $G$. Pavez-Sign\'e, Piga and Sanhueza-Matamala proved that for any function $n\leq…