KAIST Discrete Math Seminar

An L(j,k) labeling of a graph is a vertex labeling such that the difference of the labels of any two adjacent vertices is at least j and that of any two vertices of distance 2 is at least k. The minimum of the spans of all L(j,k)-labelings of G is denoted by . Recently Haque and Jha proved if G is a direct product of complete graphs, then coincide with the trivial lower bound $(N-1)k$ where N is the order of G when j/k is within a certain bound.

In this paper, we suggest a new labeling method of such a graph G. With this method, we extend the range of j/k such that holds. Moreover, we obtain an upper bound of for the remaining cases.

A Turán type problem looks for the maximum number of edges of a graph of a given order that does not contain given forms of graphs as subgraphs. These given forms of subgraphs are called the forbidden graphs of the problem. This problem also looks for the forms of graphs which have the maximum number of edges. This line of research is started by Turán who solved the problem forbidding complete graphs in 1941. Problems forbidding complete bipartite graphs or even cycles have been considered. Also there are asymptotic results on problems forbidding complete multipartite graphs. In 1994, Brualdi and Mellendorf raised the following problem which forbids a complete graph and a complete bipartite graph together.

Let t and n be positive integers with n≥t≥2. Determine the maximum number of edges of a graph of order n that contains neither K_t nor K_t,t as a subgraph.

They also solved this problem when n=2t. Obviously when n<2t, the problem reduces to Turán's problem which forbids only K_t. We solve the problem when n=2t+1. We raise a problem which forbids K_t and K_t,n-t together and a problem which forbids K_t and K_t,t,t together.