A Combinatorial Theorem on Ordered Circular Sequences of n 1 u's and n 2 v' s with Application to Kernel-perfect Graphs

Abstract

An ordered circular permutation S of u's and v' s is called an ordered circular sequence of u' s and v' s. A kernel of a digraph G=(V,A) is an independent subset of V, say K, such that for any vertex v i in V\K there is an arc from v i to a vertex v j in K. G is said to be kernel-perfect (KP) if every induced subgraph of G has a kernel. G is said to be kernel-perfect-critical (KPC) if G has no kernel but every proper induced subgraph of G has a kernel. The digraph $$ G = {\left( {V,A} \right)} = \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{C}_{n} {\left( {j_{1} ,j_{2} , \cdots ,j_{k} } \right)} $$ is defined by: V(G)={0,1,...,n−1}, A(G)={uv | v−u≡j, (mod n) for 1 ≤i≤k}. In an earlier work, we investigated the digraph $$ G = \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{C}_{n} {\left( {1, \pm \delta d, \pm 2d, \pm 3d, \cdots , \pm sd} \right)} $$ , denoted by G(n,d,r,s), where δ=1 for d>1 or δ=0 for d=1, and n,d,r,s are positive integers with (n,d)=r and n=mr, and gave some necessary and sufficient conditions for G(n,d,r,s) with r≥3 and s=1 to be KP or KPC. In this paper, we prove a combinatorial theorem on ordered circular sequences of n 1 u's and n 2 v's. By using the theorem, we prove that, if (n,d)=r≥2 and s≥2, then G(n,d,r,s) is a KP graph.