A Min-max Relation for Colourings of a Graph. Pa Perfection

This paper examines extensions of a min-max equality (stated in ? Berge, Part I) for the maximum number of nodes in a perfect graph which can be g-coloure&L A system L of linear inequalities in the variables 5 is called TDI if for every linear function cg such that _c is all integers, the dual of the linear program: maximize (~8: 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 _I sx su is TDI. It is a corollary of work of FuIkerson and _ Lovasz that: where A is a O-l matrix with no all-0 column and with the l-columns of any row not a proper subset of the l-columns of any other row, the system L(G) = {Ax s 1, x 2 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.