A generalization of perfect graphs - i-perfect graphs

Let i be a positive integer. We generalize the chromatic number x(G) of G and the clique number w(G) of G as follows: The i-chromatic number of G, denoted by xZ(G) , is the least number k for which G has a vertex partition V,, V,, . . . , Vk: such that the clique number of the subgraph induced by each V,, 1 5 j 5 k, is at most i. The i-clique numbel; denoted by w,(G), is the i-chromatic number of a largest clique in G, which equals [w(G)/ i] . Clearly x1 (G) = x(G) and w1 (G) = w(G). An induced subgraph G’ of G is an i-transversal iff w(G’) = i and w(G-G’) = w(G)i. We generalize the notion of perfect graphs as follows: (1 1 A graph G is z-pegect iff x 2 ( H ) = w Z ( H ) for every induced subgraph H of G. (2) A graph G isperfectly i-trunsversable iff either w(G) 5 i or every induced subgraph H of G with w ( H ) > i contains an i-transversal of H. We study the relationships among i-perfect graphs and perfectly i-transversable graphs. In particular, we show that 1 -perfect graphs and perfectly 1 -transversable graphs both coincide with perfect graphs, and that perfectly i-transversable graphs form a strict subset of i-perfect graphs for every i 2 2. We also show that all planar graphs are iperfect for every i 2 2 and perfectly i-transversable for every i 2 3; the latter implies *Current address: Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong. Journal of Graph Theory Vol. 23, No. 1 , 87-103 (1996)