Sharper Gaussian Covers in the Bansal--Huang--Lee Coloring Framework

Bansal, Huang, and Lee recently gave a polynomial-time algorithm that colors every $3$-colorable graph on $n$ vertices with $O(n^{0.19539})$ colors. We improve their bound to $O(n^{0.17794})$ colors. The improvement comes from a sharper rule for combining Gaussian covers. We prove the rule from Ehrhard’s inequality by comparing the probability that a Gaussian vector misses a polyhedron at two thresholds. At the farther threshold a union bound records how many halfspaces define the polyhedron, and this information reduces the loss in the combination. We then organize the successive neighborhood steps of the Bansal-Huang-Lee argument into a single recursion. The smaller loss lets the recursion run one step farther than before, and at that step it would require more distinct vertices than the graph contains. This rules out the remaining case. Combining the resulting bounded-degree guarantee with the dense-graph algorithm of Kawarabayashi, Thorup, and Yoneda gives a randomized polynomial-time algorithm that colors every $3$-colorable graph on $n$ vertices with $O(n^{0.17794})$ colors.