For every $n$, we construct a 2-dimensional $n$-element poset whose Hasse diagram has independence number $n\exp(-\Omega(\sqrt{\log n}))$, and consequently, chromatic number $\exp(\Omega(\sqrt{\log n}))$. This also yields an $n$-point planar set whose box-Delaunay graph satisfies the same bounds.
Let $G$ be the line graph of a finite simple graph, with $n\geq1$ vertices and maximum degree $\Delta$. We prove that single-site Glauber dynamics for uniform proper $q$-colorings mixes in $O_\Delta(n\log(n/\varepsilon))$ steps for every integer $q\geq\Delta+5$. Our proof uses the Bochner framework of Chen and Liu (202...
For every real $t>0$, we prove that every $2$-connected $t$-tough graph contains a cycle of length at least $\ell$ whenever $n \leq \ell(1 + 1/t)^{\lfloor \ell/2\rfloor - 1}$. This establishes the logarithmic bound conjectured by Broersma, van den Heuvel, Jung, and Veldman.
We prove that every $n$-point set in the plane has a two-coloring with discrepancy $O(n^{1/4})$ for all triangles. This improves the previous bound $O(n^{1/4}\sqrt{\log n})$ and matches the classical lower bound.
We prove that for every integers $d \ge 1$ and $s\ge2$ there exists an integer $D$, that depends only on $d$ and $s$, such that for every graph $P$ with maximum degree at most $ d$, there is a graph $H$ with maximum degree at most $D$ in which every $s$-coloring of $V(H)$ yields a monochromatic subdivision of $P$.
We extend the conjecture of Erd\H{o}s and Gy\'arf\'as on monochromatic path covers to the setting of monochromatic cycle covers. We prove that, for all $n$, every 2-edge-coloring of the complete graph on $n$ vertices contains a collection of at most $\lceil\sqrt{n}\rceil$ monochromatic cycles, all of the same color, th...
We prove that a divisor graph on an interval of integers is a planar graph whenever the interval $[n,m]$ satisfies $m\leq 7n$, and that 7 is the greatest such real number for which we can guarantee that the graph is planar. We prove these graphs are 3-colorable.
Michael D. Walton, Z. Walton· 0 citations
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