Discrete Mathematics Seminar
Spring 2025
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Fall 2024
November 1 | D. Cranston, Virginia Commonwealth University | Proper Conflict-free Coloring of Graphs with Large Maximum Degree |
Abstract:
A proper coloring of a graph is \emph{conflict-free} if, for every non-isolated vertex, some color
is used exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski proved that every
graph $G$ has a proper conflict-free coloring with at most $5\Delta(G)/2$ colors and conjectured that
$\Delta(G)+1$ colors suffice for every connected graph $G$ with $\Delta(G)\ge 3$. Our first main result
is that even for list-coloring, $\left\lceil1.6550826\Delta(G)+\sqrt{\Delta(G)}\right\rceil$ colors suffice for
every graph $G$ with $\Delta(G)\ge 10^{8}$; we also prove slightly weaker bounds for all graphs with
$\Delta(G)\ge 750$. These results follow from our more general framework on proper conflict-free
list-coloring of a pair consisting of a graph $G$ and a ``conflict'' hypergraph $H$. Our proof uses a fairly
new type of recursive counting argument called Rosenfeld counting, which is a variant of the Lov\'{a}sz Local
Lemma or entropy compression. This is joint work with Chun-Hung Liu.
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