Related papers: Line game-perfect graphs

An injective $k$-edge-coloring of a graph $G$ is a mapping $\phi$: $E(G)\rightarrow\{1,2,...,k\}$, such that $\phi(e)\ne\phi(e')$ if edges $e$ and $e'$ are at distance two, or are in a triangle. The smallest integer $k$ such that $G$ has an…

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…

The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing…

We investigate the extent to which the $k$-coloring graph $\mathcal{C}_{k}(G)$ uniquely determines the base graph $G$ and the number of colors $k$. The vertices of $\mathcal{C}_{k}(G)$ are the proper $k$-colorings of $G$, and edges connect…

A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every induced subgraph) in which at most two colors…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors…

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…

This work is concerned with the study of the Game of Graph Nim -- a class of two-player combinatorial games -- on graphs with $4$ edges. To each edge of such a graph is assigned a positive-integer-valued edge-weight, and during each round…

Consider a graph whose vertices are colored in one of two colors, say black or white. A white vertex is called integrated if it has at least as many black neighbors as white neighbors, and similarly for a black vertex. The coloring as a…

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…

For a simple graph G = (V, E) and a positive integer k greater than or equal to 2, a coloring of vertices of G using exactly k colors such that each vertex has an equal number of neighbors of each color is called neighborhood-balanced…

A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic…

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ 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…

The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours…

A signed graph is a pair $(G,\sigma)$, where $G$ is a graph and $\sigma: E(G)\rightarrow \{-, +\}$, called signature, is an assignment of signs to the edges. Given a signed graph $(G,\sigma)$ with no negative loops, a balanced…

The network coloring game has been proposed in the literature of social sciences as a model for conflict-resolution circumstances. The players of the game are the vertices of a graph with $n$ vertices and maximum degree $\Delta$. The game…

Let $H=(V(H),E(H))$ be a graph. A $k$-coloring of $H$ is a mapping $\pi : V(H) \longrightarrow \{1,2,\ldots, k\}$ so that each color class induces a $K_2$-free subgraph. For a graph $G$ of order at least $2$, a $G$-free $k$-coloring of $H$…

For given graph $H$ and graphical property $P$, the conditional chromatic number $\chi(H,P)$ of $H$, is the smallest number $k$, so that $V(H)$ can be decomposed into sets $V_1,V_2,\ldots, V_k$, in which $H[V_i]$ satisfies the property $P$,…

A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a \emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…

Consider the balanced Ramsey game, in which a player has r colors and where in each step r random edges of an initially empty graph on n vertices are presented. The player has to immediately assign a different color to each edge and her…