Linear inequalities among graph invariants: Using GraPHedron to uncover optimal relationships

Abstract

Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a "small" number of vertices. It greatly helps in conjecturing optimal linear inequalities, which are then hopefully proved for any number of vertices. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities for the stability number and the number of edges are obtained. To this aim, a problem of Ore (1962) related to the Turán Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter.

PHOEG Helps to Obtain Extremal Graphsch:32

“…We note that this idea has led to several results (see e.g., [15]) including a complete answer [16] to an open problem introduced by Ore in 1962 [17]. A difference from existing softwares based on metaheuristics is that these bounds are exact for the graphs used in the convex hull.…”

Fibonacci index and stability number of graphs: a polyhedral study

“…Turán graphs T n,α have minimum size inside G(n, α) by the Theorem of Turán [23]. Christophe et al [6] give a tight lower bound for the connected case of this theorem, and Bougard and Joret [2] characterized the extremal graphs, which happen to contain the TC n,α graphs as a subclass.…”

Turán's theorem and k‐connected graphs

“…Determining f (n, α, 1) was in fact an old problem of Ore [10] which has been settled recently independently by Christophe et al [3] and by Gitler and Valencia [6]. They proved the following result, where t(n, α) is the size of the Turán graph T (n, α).…”