Related papers: Totally Silver Graphs

Let $G = (V,E)$ be a finite simple undirected graph without isolated vertices. A bijective map $f: V \cup E \rightarrow \{1,2, \dots, |V|+ |E| \}$ is called total local antimagic labeling if for each edge $uv \in E, w(u) \ne w(v)$, where…

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption…

Graph colorings are becoming an increasingly useful family of mathematical models for a broad range of applications, such as time tabling and scheduling, frequency assignment, register allocation, computer security and so on. Graph proper…

A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow\{0,1,2,\ldots,k\}$ with $\sum\limits_{e\in E_G(v)}f(e)\leq k$ for each vertex $v$ of $G$, where $E_G(v)$ is the set of edges incident with $v$ in $G$. A perfect $k$-matching of…

A 2-distance k-coloring of a graph G is a mapping from V (G) to the set of colors {1,. .. , k} such that every two vertices at distance at most 2 receive distinct colors. The 2-distance chromatic number $\chi$ 2 (G) of G is then the mallest…

For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring…

The pseudo-Grundy index of a graph is the largest number of colors that can be assigned to its edges, such that for every pair of colors $i,j$, if $i < j$ then every edge colored with color $j$ is adjacent to at least one edge colored with…

It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained…

A vertex coloring $\varphi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$, either $\varphi$ uses more than $p$ colors on $H$, or there is a color that appears exactly once on $H$. We prove that for every fixed…

We say a proper coloring of a graph is distance-$k$ fall if every vertex is within distance $k$ of at least one vertex of every color. We show that if $G$ is a connected graph of order at least $3$ that is $3$-colorable, thenit has a…

We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in…

Let $G$ be a graph, $r \geq t$ integers, and $N \subseteq E(G)$. An $(r,t)$-threshold-coloring of $G$ with respect to $N$ is a mapping $c: V(G) \rightarrow \{0,\ldots,r-1\}$ such that $|c(u)-c(v)| \leq t$ for every $uv \in N$ and…

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph is colored. A path in a total-colored graph is a {\it total proper path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any two…

The Colouring problem asks whether the vertices of a graph can be coloured with at most $k$ colours for a given integer $k$ in such a way that no two adjacent vertices receive the same colour. A graph is $(H_1,H_2)$-free if it has no…

A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is…

An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We…

A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex $k$-connected} if any two vertices of the graph are connected by $k$ disjoint…

A proper coloring of a graph is \emph{conflict-free} if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski proved that every graph $G$ has a proper conflict-free…

A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$…