相关论文: A note on the connected game coloring number

Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum…

A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible…

Motivated by the success of domination games and by a variation of the coloring game called the indicated coloring game, we introduce a version of domination games called the indicated domination game. It is played on an arbitrary graph $G$…

An odd coloring of a graph $G$ is a proper vertex coloring $\varphi$ with the property that for each non-isolated vertex $v\in V(G)$, there exists a color $c$ such that the cardinality of $\varphi^{-1}(c)\cap N(v)$ is odd. The concept of…

The idea that those different from you are "unfriendly" is captured in the definition of unfriendly 2-colorings in graph theory in a paper by Aharoni, Milner and Prikry, where they prove that every finite graph has an unfriendly coloring.…

Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if…

A $k$-coloring of a tournament is a partition of its vertices into $k$ acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a…

An ordering of the vertices of a graph is \emph{connected} if every vertex (but the first) has a neighbor among its predecessors. The greedy colouring algorithm of a graph with a connected order consists in taking the vertices in order, and…

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in…

Hoffman's bound is a well-known spectral bound on the chromatic number of a graph, known to be tight for instance for bipartite graphs. While Hoffman colorings (colorings attaining the bound) were studied before for regular graphs, for…

Zero forcing is a dynamic graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. This forcing process has been used to approximate certain linear algebraic parameters, as well as…

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. In this paper, we completely determine the edge chromatic number of outer 1-planar graphs.

We introduce a class of cooperative games induced by weighted directed graphs. Specifically, the coalitional value combines an internal interaction term given by the induced subgraph game with an external component based on minimal incoming…

The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. The DP-chromatic number of a graph $G$, $\chi_{_{DP}}(G)$, is the analogue of the chromatic number of…

We show that there exists a constant $c > 0$ such that if $G$ is a planar graph with 5-correspondence assignment $(L,M)$, then $G$ has at least $2^{c\cdot v(G)}$ distinct $(L,M)$-colourings. This confirms a conjecture of Langhede and…

An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected}…

DP-coloring was introduced by Dvo\v{r}\'{a}k and Postle as a generalization of list coloring and signed coloring. A new coloring, strictly $f$-degenerate transversal, is a further generalization of DP-coloring and $L$-forested-coloring. In…

Low-treedepth colorings are an important tool for algorithms that exploit structure in classes of bounded expansion; they guarantee subgraphs that use few colors have bounded treedepth. These colorings have an implicit tradeoff between the…

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…