A min-max relation for the partial q- colourings of a graph. Part II: Box perfection

Abstract

This paper examines extensions of a min-max equality (stated in C Berge, Part I) for the maximum number of nodes in a perfect graph which can be q-coloured. A system L of linear inequalities in the variables x is called TDI if for every linear function c x such that c is all integers, the dual of the linear program: maximize { c x : x satisfies L} has an integer-valued optimum solution or no optimum solution. A system L is called box TDI if L together with any inequalities l⩽ x⩽ u is TDI. It is a corollary of work of Fulkerson and Lov́asz that: where A is a 0–1 matrix with no all-0 column and with the 1-columns of any row not a proper subset of the 1-columns of any other row, the system L(G) = {A x⩽ 1, x⩾ 0} is TDI if and only if A is the matrix of maximal cliques (rows) versus nodes (columns) of a perfect graph. Here we will describe a class of graphs in a graph-theoretic way, and characterize them as the graphs G for which the system L( G) is box TDI. Thus we call these graphs box perfect. We also describe some classes of box perfect graphs.