Related papers: Injective edge-coloring of graphs with small maxim…
The semistrong edge coloring, as a relaxation of the well-known strong edge coloring, can be used to model efficient communication scheduling in wireless networks. An edge coloring of a graph $G$ is called \emph{semistrong} if every color…
The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star…
The chromatic edge-stability number ${\rm es}_{\chi}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph $G'$ with $\chi(G')=\chi(G)-1$. Edge-stability critical graphs are introduced as the graphs…
The star chromatic index of a graph $G$, denoted by $\chi^\prime_s(G)$, is the smallest integer $k$ for which $G$ admits a proper edge coloring with $k$ colors such that every path and cycle of length four is not bicolored. Let $d$ be the…
The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $p$ such that vertices of $G$ can be partitioned into disjoint classes $X_{1}, ..., X_{p}$ where vertices in $X_{i}$ have pairwise distance greater than…
A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of…
A $k$-proper edge-coloring of a graph G is called adjacent vertex-distinguishing if any two adjacent vertices are distinguished by the set of colors appearing in the edges incident to each vertex. The smallest value $k$ for which $G$ admits…
An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex…
Indicated coloring is a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly,…
In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer…
An \emph{edge coloring} of a graph $G$ is strong if each color class is an induced matching of $G$. The \emph{strong chromatic index} of $G$, denoted by $\chi _{s}^{\prime }(G)$, is the minimum number of colors for which $G$ has a strong…
In this paper, we introduce and study a novel graph parameter called the $k$-defect number, denoted $\phi_{k}(G)$, for a graph $G$ and an integer $0\leq k\leq |E(G)|$. Unlike traditional defective colorings that bound the local degree…
For a sequence of non-decreasing positive integers $S = (s_1, \ldots, s_k)$, a packing $S$-coloring is a partition of $V(G)$ into sets $V_1, \ldots, V_k$ such that for each $1\leq i \leq k$ the distance between any two distinct $x,y\in V_i$…
Dvo\v{r}\'ak \emph{et al.} introduced a variant of the Randi\'c index of a graph $G$, denoted by $R'(G)$, where $R'(G)=\sum_{uv\in E(G)}\frac 1 {\max\{d(u), d(v)\}}$, and $d(u)$ denotes the degree of a vertex $u$ in $G$. The coloring number…
A graph is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on the set of all vertices receiving the same colour is at most k. The k-defective chromatic number $\chi_k(G)$…
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge…
A \emph{star coloring} of a graph $G$ is a proper vertex-coloring such that no path on four vertices is $2$-colored. The minimum number of colors required to obtain a star coloring of a graph $G$ is called star chromatic number and it is…
Given an edge-coloring of a simple graph, assign to every vertex $v$ a set $S_v$ comprised of the colors used on the edges incident to $v$. The $k$-intersection chromatic index of a graph is the minimum $t$ such that the edge set can be…
A proper coloring of the vertices of a graph is called a \emph{star coloring} if the union of every two color classes induces a star forest. The star chromatic number $\chi_s(G)$ is the smallest number of colors required to obtain a star…
A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. It is known that the list edge chromatic number $\chi'_l(G)$ of any outer-1-planar graph $G$ with…