Results 1 
8 of
8
On the bchromatic number of regular graphs
, 2010
"... The bchromatic number of a graph G is the largest integer k such that G admits a proper kcoloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every dregular graph with at least 2d3 vertices has bchromatic number d+ ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The bchromatic number of a graph G is the largest integer k such that G admits a proper kcoloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every dregular graph with at least 2d3 vertices has bchromatic number d+1, that the bchromatic number of an arbitrary dregular graph with girth g = 5 is at least ⌊ ⌋ d+1 2 and that every dregular graph, d ≥ 6, with diameter at least d and with no 4cycles admits a bcoloring with d+1 colors.
On the bicoloring of cographs and . . .
"... A bcoloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The bchromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a bcoloring with t colors. A graph G ..."
Abstract
 Add to MetaCart
A bcoloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The bchromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a bcoloring with t colors. A graph G is bcontinuous if it admits a bcoloring with t colors, for every t = χ(G),..., χb(G). We define a graph G to be bmonotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. In this work, we prove that P4sparse graphs (and, in particular, cographs) are bcontinuous and bmonotonic. Besides, we describe a dynamic programming algorithm to compute the bchromatic number in polynomial time within these graph classes.
The bchromatic index of a graph
, 2012
"... The bchromatic index ϕ′(G) of a graph G is the largest integer k such that G admits a proper kedge coloring in which every color class contains at least one edge incident to some edge in all the other color classes. The bchromatic index of trees is determined and equals either to a natural upper ..."
Abstract
 Add to MetaCart
The bchromatic index ϕ′(G) of a graph G is the largest integer k such that G admits a proper kedge coloring in which every color class contains at least one edge incident to some edge in all the other color classes. The bchromatic index of trees is determined and equals either to a natural upper bound m ′ (T) or one less, where m ′ (T) is connected with the number of edges of high degree. Some conditions are given for which graphs have the bchromatic index strictly less than m ′ (G), and for which conditions it is exactly m ′ (G). In the last part of the paper regular graphs are considered. It is proved that with four exceptions, the bchromatic number of cubic graphs is 5. The exceptions are K4, K3,3, the prism over K3, and the cube Q3.
On the bcoloring of P4tidy graphs
, 2010
"... A bcoloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The bchromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a bcoloring with t colors. A graph G i ..."
Abstract
 Add to MetaCart
A bcoloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The bchromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a bcoloring with t colors. A graph G is bcontinuous if it admits a bcoloring with t colors, for every t = χ(G),..., χb(G), and it is bmonotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. In this work, we prove that P4tidy graphs (a generalization of many classes of graphs with few induced P4s) are bcontinuous and bmonotonic. Furthermore, we describe a polynomial time algorithm to compute the bchromatic number for this class of graphs.
Bounds for Minimum Feedback Vertex Sets in Distance Graphs and Circulant Graphs
"... For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with i − j  ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of v ..."
Abstract
 Add to MetaCart
For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with i − j  ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.
On the bchromatic number of some graph products
, 2011
"... A bcoloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the bchromatic number is the largest integer ϕ(G) for which a graph has a bcoloring with ϕ(G) colors. We determine some upper and lower bounds for the ..."
Abstract
 Add to MetaCart
A bcoloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the bchromatic number is the largest integer ϕ(G) for which a graph has a bcoloring with ϕ(G) colors. We determine some upper and lower bounds for the bchromatic number of the strong product G H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the