Rafael R. Kamalian

On complexity of special maximum matchings constructing

For bipartite graphs the NP-completeness is proved for the problem of existence of maximum matching which removal leads to a graph with given lower(upper) bound for the cardinality of its maximum matching.

On sum edge-coloring of regular, bipartite and split graphs

An edge-coloring of a graph G with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of G are distinct and the sum of the colors of the edges of G is minimum. The edge-chromatic sum of a graph G is the sum of the colors of edges in a sum edge-coloring of G. It is known that the problem of finding the edge-chromatic sum of an r-regular (r>=3) graph is NP-complete. In this paper we give a polynomial time (1+2r(r+1)^2)-approximation algorithm for the edge-chromatic sum problem on r-regular graphs for r>=3. Also, it is known that the problem of finding the edge-chromatic sum of bipartite graphs with maximum degree 3 is NP-complete. We show that the problem remains NP-complete even for some restricted class of bipartite graphs with maximum degree 3. Finally, we give upper bounds for the edge-chromatic sum of some split graphs.