The b-chromatic number (G) of a graph G is defined as the largest number k for which the vertices of G can be colored with k colors satisfying the following property: for each i, 1 i k, there exists a vertex x i of color i such that for all j = i, 1 j k there exists a vertex y j of color j adjacent tois the chromatic number of H. We characterize all b-perfect bipartite graphs and all b-perfect P 4 -sparse graphs by minimal forbidden induced subgraphs. We also prove that every 2K 2 -free and P 5 -free graph is b-perfect.
In this paper we obtain some upper bounds for b-chromatic number of K 1,t -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight.
Kotzig asked in 1979 what are necessary and sufficient conditions for a d-regular simple graph to admit a decomposition into paths of length d for odd d>3. For cubic graphs, the existence of a 1-factor is both necessary and sufficient. Even more, each 1-factor is extendable to a decomposition of the graph into paths of length 3 where the middle edges of the paths coincide with the 1-factor. We conjecture that existence of a 1-factor is indeed a sufficient condition for Kotzig's problem. For general odd regular graphs, most 1-factors appear to be extendable and we show that for the family of simple 5-regular graphs with no cycles of length 4, all 1-factors are extendable. However, for d>3 we found infinite families of d-regular simple graphs with non-extendable 1-factors. Few authors have studied the decompositions of general regular graphs. We present examples and open problems; in particular, we conjecture that in planar 5-regular graphs all 1-factors are extendable. ᭧
The mean distance of a simple connected graph G of order n is defined by z -dfx'y) . We answer a problem of Plesnik about the edge-vulnerability of G where e is an related to p , i.e., we find a bound for p(G -e) -p(G) and for edge of G such that G -e is still connected. This question is of interest in interconnection
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