Graph coloring is the mathematical model for studying problems related to conflict-free allocation of resources. DP-coloring (also known as... Show moreGraph coloring is the mathematical model for studying problems related to conflict-free allocation of resources. DP-coloring (also known as correspondence coloring) of graphs is a vast generalization of classic graph coloring, and many more concepts of colorings studied in the past 150+ years.
We study problems in DP-coloring of graphs that combine questions and ideas from extremal, structural, probabilistic, and enumerative aspects of graph coloring. In particular, we study (i) DP-coloring Cartesian products of graphs using the DP-color function, the DP coloring counterpart of the Chromatic polynomial, and robust criticality, a new notion of graph criticality; (ii) Shameful conjecture on the mean number of colors used in a graph coloring, in the context of list coloring and DP-coloring; and (iii) asymptotic bounds on the difference between the chromatic polynomial and the DP color function, as well as the difference between the dual DP color function and the chromatic polynomial, in terms of the cycle structure of a graph. These results respectively give an upper bound and a lower bound on the chromatic polynomial in terms of DP colorings of a graph. Show less
