Graphs and Matroids- Tom Kelly

Title: Fractional coloring with local demands

Abstract: In a fractional coloring, vertices of a graph are assigned subsets of the $[0, 1]$-interval such that adjacent vertices receive disjoint subsets. The fractional chromatic number of a graph is at most $k$ if it admits a fractional coloring in which the amount of ``color'' assigned to each vertex is at least $1/k$.  We investigate fractional colorings where vertices ``demand'' different amounts of color, determined by local parameters such as the degree of a vertex.  Many well-known results concerning the fractional chromatic number and independence number have natural generalizations in this new paradigm.  We discuss several such results as well as open problems.  In particular, we will sketch a proof of a ``local demands'' version of Brooks' Theorem that considerably generalizes the Caro-Wei Theorem and implies new bounds on the independence number.  Joint work with Luke Postle.