Given a graph G , an r -hued coloring of G is a proper vertex coloring such that for every vertex v , the number of colors appearing in its neighborhood is at least min { d G ( v ) , r } , where d G ( v ) denotes the degree of v in G . The r -hued chromatic number χ r ( G ) is the smallest number of colors needed for an r -hued coloring of G . In this paper, we study the r -hued coloring of complements of cycles. For any positive integer r , we completely determine the r -hued chromatic number of C n by constructing explicit coloring schemes and analyzing the sizes of independent sets of color classes.
An equitable k-coloring is a proper coloring with color classes V_1,V_2,\ldots,V_{k} for which the numbers of vertices in any two color classes differ by at most one. A graph G is equitably k-colorable if there is an equitable k-coloring. The smallest positive integer k such that G is equitably k-colorable is the equit...
Annob Jobpan, K. Nakprasit, K. Nakprasit· International Journal of Mat...· 0 citations
The dominated coloring of a graph $G$ is a proper vertex coloring such that each color class is dominated by a vertex. The minimum number of colors required for a dominated coloring of a graph $G$ is called the dominated chromatic number of the graph $G$ and is denoted by $\chi_{dom}(G)$. A graph $G$ is said to be chro...
We introduce and begin the study of sequence b-colorings, a natural generalization of the classical notion of b-colorings introduced by Irving and Manlove in 1999. In a sequence b-coloring, each color class is required to contain a prescribed minimum number of color-dominating vertices (CDVs). We establish several fund...
An acyclic edge coloring of a graph G is a proper edge coloring such that G contains no bichromatic cycles. The acyclic chromatic index χa′(G) is the minimum number of colors required for an acyclic edge coloring. Fiamčik and Alon et al. independently conjectured that χa′(G)≤Δ+2 for every simple graph G with maximum de...
We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its ori...
Tomohiro Koana, Y. Oh, Hirotaka Yoneda· 0 citations
We prove the conjecture of Borowiecki and Broere that every finite simple planar graph admits a generalized total coloring with four colors: a proper vertex coloring with four colors together with an edge coloring with four colors in which each edge color class is a forest and no edge receives the color of either endpo...
James Alexander Schreib· 0 citations
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