Distribution of colors in Gallai colorings

A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1\ge e_2 \ge \dots \ge e_k$ with $\sum_{i=1}^ke_i={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$? In this paper, we give several sufficient conditions and one necessary condition to guarantee a positive answer to the above question. In particular, we prove the existence of a Gallai-coloring if $e_1-e_k\le 1$ and $k \le \lfloor n/2\rfloor$. We prove that for any integer $k\ge 3$ there is a (unique) integer $g(k)$ with the following property: there exists a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$ for every $e_1\le\dots \le e_k$ satisfying $\sum_{i=1}^ke_i={n\choose 2}$, if and only if $n\ge g(k)$. We show that $g(3)=5$, $g(4)=8$, and $2k-2\le g(k)\le 8k^2+1$ for every $k\ge 3$

Transitive and Gallai colorings

A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of comparability graphs and anti-Ramsey theory. We introduce a transitive analogue for acyclic directed graphs, and generalize both notions to Coxeter systems, matroids and commutative algebras. It is shown that for any finite matroid (or oriented matroid), the maximal number of colors is equal to the matroid rank. This generalizes a result of Erd\H{o}s-Simonovits-S\'os for complete graphs. The number of Gallai (or transitive) colorings of the matroid that use at most $k$ colors is a polynomial in $k$. Also, for any acyclic oriented matroid, represented over the real numbers, the number of transitive colorings using at most 2 colors is equal to the number of chambers in the dual hyperplane arrangement. We count Gallai and transitive colorings of the root system of type A using the maximal number of colors, and show that, when equipped with a natural descent set map, the resulting quasisymmetric function is symmetric and Schur-positive.Comment: 31 pages, 5 figure

Improved bounds on the multicolor Ramsey numbers of paths and even cycles

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest

Flexibility of graphs with maximum average degree less than 33

In the flexible list coloring problem, we consider a graph $G$ and a color list assignment $L$ on $G$, as well as a set of coloring preferences at some vertex subset of $G$. Our goal is to find a proper $L$-coloring of $G$ that satisfies some given proportion of these coloring preferences. We say that $G$ is $\epsilon$-flexibly $k$-choosable if for every $k$-size list assignment $L$ on $G$ and every set of coloring preferences, $G$ has a proper $L$-coloring that satisfies an $\epsilon$ proportion of these coloring preferences. Dvo\v{r}\'ak, Norin, and Postle [Journal of Graph Theory, 2019] asked whether every $d$-degenerate graph is $\epsilon$-flexibly $(d+1)$-choosable for some constant $\epsilon = \epsilon(d) > 0$. In this paper, we prove that there exists a constant $\epsilon > 0$ such that every graph with maximum average degree less than $3$ is $\epsilon$-flexibly $3$-choosable, which gives a large class of $2$-degenerate graphs which are $\epsilon$-flexibly $(d+1)$-choosable. In particular, our results imply a theorem of Dvo\v{r}\'ak, Masa\v{r}\'ik, Mus\'ilek, and Pangr\'ac [Journal of Graph Theory, 2020] stating that every planar graph of girth $6$ is $\epsilon$-flexibly $3$-choosable for some constant $\epsilon > 0$. To prove our result, we generalize the existing reducible subgraph framework traditionally used for flexible list coloring to allow reducible subgraphs of arbitrarily large order.Comment: 22 page