LIST COLORING OF BLOCK GRAPHS AND COMPLETE BIPARTITE GRAPHS

List coloring is a vertex coloring of a graph where each vertex can be restricted to a list of allowed colors. For a given graph G and a set L(v) of colors for every vertex v, a list coloring is a function that maps every vertex v to a color in the list L(v) such that no two adjacent vertices receive the same color. It was first studied in the 1970s in independent papers by Vizing and by Erdős, Rubin, and Taylor. A block graph is a type of undirected graph in which every biconnected component (block) is a clique. A complete bipartite graph is a bipartite graph with partitions V 1, V 2 such that for every two vertices v_1∈V_1 and v_2∈V_2 there is an edge (v 1, v 2). If |V_1 |=n and |V_2 |=m it is denoted by K_(n,m). In this paper we provide a polynomial algorithm for finding a list coloring of block graphs and prove that the problem of finding a list coloring of K_(n,m) is NP-complete even if for each vertex v the length of the list is not greater than 3 (|L(v)|≤3).

Common Independence in Graphs

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

POWERS OF BINOMIAL EDGE IDEALS WITH QUADRATIC GRÖBNER BASES

We study powers of binomial edge ideals associated with closed and block graphs.