Fractional coloring with local demands and applications to degree-sequence bounds on the independence number

Abstract

In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most $k$ if it has a fractional coloring in which each vertex receives a subset of $[0, 1]$ of measure at least $1/k$. We introduce and develop the theory of "fractional colorings with local demands" wherein each vertex "demands" a certain amount of color that is determined by local parameters such as its degree or the clique number of its neighborhood. This framework provides the natural setting in which to generalize degree-sequence type bounds on the independence number. Indeed, by Linear Programming Duality, all of the problems we study have an equivalent formulation as a problem concerning weighted independence numbers, and they often imply new bounds on the independence number. Our results and conjectures are inspired by many of the most classical results and important open problems concerning the independence number and the chromatic number, often simultaneously. We conjecture a local strengthening of both Shearer's bound on the independence number of triangle-free graphs and the fractional relaxation of Molloy's recent bound on their chromatic number, as well as a longstanding problem of Ajtai et al.\ on the independence number of $K_r$-free graphs and the fractional relaxations of Reed's $ω, Δ, χ$ Conjecture and the Total Coloring Conjecture. We prove an approximate version of the first two, and we prove "local demands" versions of Vizing's Theorem and of some $χ$-boundedness results.