Реберный Ck-граф графа

For any integer k≥4, the edge Ck graph Ek(G) of a graph G=(V,E) has all edges of G as it vertices, two vertices in Ek(G) are adjacent if their corresponding edges in G are either incident or belongs to a copy of Ck. In this paper, we obtained the characterizations for the edge Ck graph of a graph G to be connected, complete, bipartite etc. It is also proved that the edge C4 graph has no forbidden subgraph characterization. Mereover, the dynamical behavior such as convergence, periodicity, mortality and touching number of Ek(G) are studied.

SUB-COLORING AND HYPO-COLORING INTERVAL GRAPHS

In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as "subroutines" for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a "forbidden subgraph" characterization of graphs with sub-chromatic number k and use this to derive a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete, and then via reduction to the max-coloring problem, show how to obtain an O( log n)-approximation algorithm for it.