Related papers: The absence of monochromatic triangle implies vari…

Let $G$ be an edge-colored graph with $n$ vertices. A subgraph $H$ of $G$ is called a rainbow subgraph of $G$ if the colors of each pair of the edges in $E(H)$ are distinct. We define the minimum color degree of $G$ to be the smallest…

An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomass\'e, we prove that the vertex set of any $2$-locally…

For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a graph, that is, assignments of $k$ distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly…

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this…

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 {\em total coloring} of a graph $G$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent/incident elements receive distinct colors. The {\em total chromatic number} of a graph $G$, denoted by…

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…

An edge-colored graph is a graph in which each edge is assigned a color. Such a graph is called strongly edge-colored if each color class forms an induced matching, and called rainbow if all edges receive pairwise distinct colors. In this…

We prove that for every tree $T$ which is not an edge, for almost every graph $G$ which does not contain $T$ as an induced subgraph, $V(G)$ has a partition into $\alpha(T)-1$ parts certifying this fact. Each part induces a graph which is…

Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the…

Let $G$ be an edge-colored graph, a walk in $G$ is said to be a properly colored walk iff each pair of consecutive edges have different colors, including the first and the last edges in case that the walk be closed. Let $H$ be a graph…

Let $G$ be a nontrivial connected graph of order $n$ with an edge-coloring $c:E(G)\rightarrow\{1,2,\dots,t\}$,$t\in\mathbb{N}$, where adjacent edges may be colored with the same color. A tree $T$ in $G$ is a \emph{proper tree} if no two…

A tree $T$ in an edge-colored graph is a \emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2\leq k\leq n$. For a vertex set $S\subseteq…

Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly…

An edge-colored graph is said to be balanced if it has an equal number of edges of each color. Given a graph $G$ whose edges are colored using two colors and a positive integer $k$, the objective in the Edge Balanced Connected Subgraph…

A repetition is a sequence of symbols in which the first half is the same as the second half. An edge-coloring of a graph is repetition-free or nonrepetitive if there is no path with a color pattern that is a repetition. The minimum number…

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors needed for a…

For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of…

A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least…

A graph $G$ is conflict-free $k$-edge-colorable if there exists an assignment of $k$ colors to $E(G)$ such that for every edge $e\in E(G)$, there is a color that is assigned to exactly one edge among the closed neighborhood of $e$. The…