A vertex 2-coloring of a graph is said to be perfect with parameters (aij) k i,j=1 if for every i, j ∈ {1, ..., k} every vertex of color i is adjacent with exactly i vertices of color j. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube {0, 1}24 with parameters ((20+ c, 256− c)(c, 276− c)) (i.e., with eigenvalue 20). We prove that such colorings exist if c = 3, 6, 8, 9,...

In this paper we consider the following type of game: two players must color the vertices of a given graph G = (V, E), in a prescribed order, in such a way that no two adjacent vertices are colored with the same color. In one variant, the first player which is unable to move loses the game. In another variant, player 1 wins the game, if and only if the game ends with all vertices colored. In th...

We present an O(n) algorithm that computes a maximum stable set of any perfect graph with no balanced skew-partition. We present O(n) time algorithm that colors them.

A perfect coloring (equivalent concepts are equitable partition and design) of a graph $G$ is function $f$ from the set vertices onto some finite (of colors) such that every node color $i$ has exactly $S(i,j)$ neighbors $j$, where constants, forming matrix $S$ called quotient. If an adjacency simple $T$ on colors, then covering target by cover $G$. We characterize all coverings infinite square ...

A coloring of a tree is convex if the vertices that pertain to any color induce a connected subtree; a partial coloring (which assigns colors to some of the vertices) is convex if it can be completed to a convex (total) coloring. Convex colorings of trees arise in areas such as phylogenetics, linguistics, etc., e.g., a perfect phylogenetic tree is one in which the states of each character induc...

Coloring graphs is one of important and frequently used topics in diverse sciences. In the majority of the articles, it is intended to find a proper bound for vertex coloring, edge coloring or total coloring in the graph. Although it is important to find a proper algorithm for graph coloring, it is hard and time-consuming too. In this paper, a new algorithm for vertex coloring, edge coloring an...

An r-component connected coloring of a graph is a coloring of the vertices so that each color class induces a subgraph having at most r connected components. The concept has been well-studied for r = 1, in the case of trees, under the rubric of convex coloring , used in modeling perfect phylogenies. Several applications in bioinformatics of connected coloring problems on general graphs are disc...

The binary perfect phylogeny model is too restrictive to model biological events such as back mutations. In this paper we consider a natural generalization of the model that allows a special type of back mutation. We investigate the problem of reconstructing a near perfect phylogeny over a binary set of characters where characters are persistent: characters can be gained and lost at most once. ...

We call a graph 2K2-free if it is connected and does not contain two independent edges as an induced subgraph. The assumption of connectedness in this definition only serves to eliminate isolated vertices. Wagon [6] proved that x(G) ~ w(G)[w(G) + 1]/2 if G is 2Krfree where x(G) and w(G) denote respectively the chromatic number and maximum clique size of G. Further properties of 2K2-free graphs ...