Remarks on the distribution of colors in Gallai colorings

A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \cdots \ge e_k$ of positive integers is an $(n,k)$-sequence if ${\sum }_{i=1}^k{e}_i=\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. An $(n,k)$-sequence is a G-sequence if there is a Gallai coloring of $K_n$ with $k$ colors such that there are $e_i$ edges of color $i$ for all $i,1\le i\le k$. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer $k\ge 3$ there exists an integer $g\left(k\right)$ such that every $(n,k)$-sequence is a G-sequence if and only if $n\ge g\left(k\right)$. They showed that $g\left(3\right)=5,g\left(4\right)=8$ and $2k-2\le g\left(k\right)\le 8k^2+1$.We show that $g\left(5\right)=10$ and give almost matching lower and upper bounds for $g\left(k\right)$ by showing that with suitable constants $\alpha ,\beta >0$, $\frac{\alpha k^{1.5}}{lnk}\le g\left(k\right)\le \beta k^{1.5}$ for all sufficiently large $k$.     

Vertex-disjoint rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph of order $n$. The minimum color degree of $G$, denoted by ${\delta }^c\left(G\right)$, is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if ${\delta }^c\left(G\right)\ge (n+1)∕2$, then $G$ contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if $n\ge 20$ and ${\delta }^c\left(G\right)\ge (n+2)∕2$, then $G$ contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if ${\delta }^c\left(G\right)\ge (n+k)∕2$, then $G$ contains $k$ vertex-disjoint rainbow triangles. For any integer $k\ge 2$, we show that if $n\ge 16k-12$ and ${\delta }^c\left(G\right)\ge n∕2+k-1$, then $G$ contains $k$ vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of $k$ edge-disjoint rainbow triangles.     

Minimum degree condition for a graph to be knitted

For a positive integer $k$, a graph is $k$-knitted if for each subset $S$ of $k$ vertices, and every partition of $S$ into (disjoint) parts $S_1,\dots ,S_t$ for some $t\ge 1$, one can find disjoint connected subgraphs $C_1,\dots ,C_t$ such that $C_i$ contains $S_i$ for each $i\in \left[t\right]?\{1,2,\dots ,t\}$. In this article, we show that if the minimum degree of an $n$-vertex graph $G$ is at least $n∕2+k∕2?1$ when $n\ge 2k+3$, then $G$ is $k$-knitted. The minimum degree is sharp. As a corollary, we obtain that $k$-contraction-critical graphs are $?\frac{k}{8}?$-connected.     

Improvements on Hippchen's conjecture

Let G be a k-connected graph on n vertices. Hippchen's Conjecture (2008) states that two longest paths in G share at least k vertices. Gutiérrez (2020) recently proved the conjecture when $k\le 4$ or $k\ge \frac{n?2}{3}$. We improve upon both results; namely, we show that two longest paths in G share at least k vertices when $k=5$ or $k\ge \frac{n+2}{5}$. This completely resolves two conjectures by Gutiérrez in the affirmative.     

Bounds for moments of cubic and quartic Dirichlet L-functions

We study the $2k$-th moment of central values of the family of primitive cubic and quartic Dirichlet $L$-functions. We establish sharp lower bounds for all real $k\ge 1/2$ unconditionally for the cubic case and under the Lindelöf hypothesis for the quartic case. We also establish sharp lower bounds for all real $0\le k<1/2$ and sharp upper bounds for all real $k\ge 0$ for both the cubic and quartic cases under the generalized Riemann hypothesis (GRH). As an application of our results, we establish quantitative non-vanishing results for the corresponding $L$-values.