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Register a new user2-Colored Matchings in a 3-Colored K^{3}_{12}
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Let $K_{n}^{r}$ denote the complete $r$-uniform hypergraph on $n$ vertices. A matching $M$ in a hypergraph is a set of pairwise vertex disjoint edges. Recent Ramsey-type results rely on lemmas about the size of monochromatic matchings. A starting point for this study comes from a well-known result of Alon, Frankl, and Lovasz (1986). Our motivation is to find the smallest $n$ such that every $t$-coloring of $K_{n}^{r}$ contains an $s$-colored matching of size $k$. It has been conjectured that in every coloring of the edges of $K_n^r$ with 3 colors there is a 2-colored matching of size at least $k$ provided that $n geq kr + lfloor frac{k-1}{r+1} rfloor$. The smallest test case is when $r=3$ and $k=4$. We prove that in every 3-coloring of the edges of $K_{12}^3$ there is a 2-colored matching of size 4.
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There has been much research on the topic of finding a large rainbow matching (with no two edges having the same color) in a properly edge-colored graph, where a proper edge coloring is a coloring of the edge set such that no same-colored edges are incident. Barat, Gyarfas, and Sarkozy conjectured that in every proper edge coloring of a multigraph (with parallel edges allowed, but not loops) with $2q$ colors where each color appears at least $q$ times, there is always a rainbow matching of size $q$. Recently, Aharoni, Berger, Chudnovsky, Howard, and Seymour proved a relaxation of the conjecture with $3q-2$ colors. Our main result proves that $2q + o(q)$ colors are enough if the graph is simple, confirming the conjecture asymptotically for simple graphs. This question restricted to simple graphs was considered before by Aharoni and Berger. We also disprove one of their conjectures regarding the lower bound on the number of colors one needs in the conjecture of Barat, Gyarfas, and Sarkozy for the class of simple graphs. Our methods are inspired by the randomized algorithm proposed by Gao, Ramadurai, Wanless, and Wormald to find a rainbow matching of size $q$ in a graph that is properly edge-colored with $q$ colors, where each color class contains $q + o(q)$ edges. We consider a modified version of their algorithm, with which we are able to prove a generalization of their statement with a slightly better error term in $o(q)$. As a by-product of our techniques, we obtain a new asymptotic version of the Brualdi-Ryser-Stein Conjecture.
A generalized crank ($k$-crank) for $k$-colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined $k$-crank. Namely, we first obtain some inequalities between the $k$-crank counts $M_{k}(r,m,n)$ for $m=2,3$ and $4$, then we prove the positivity of symmetrized even $k$-crank moments weighted by the parity for $k=2$ and $3$. We conclude with several remarks on furthering the study initiated here.
Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-H{a}ggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers $s$ and $t$ with $tgeq sgeq2$, every edge-colored graph $G$ with no properly colored $K_{s,t}$ contains a spanning subgraph $H$ which admits an orientation $D$ such that every directed cycle in $D$ is a properly colored cycle in $G$. Using this result, we show that for $rgeq4$, if the Caccetta-H{a}ggkvist Conjecture holds , then every edge-colored graph of order $n$ with minimum color degree at least $n/r+2sqrt{n}+1$ contains a properly colored cycle of length at most $r$. In addition, we also obtain an asymptotically tight total color degree condition which ensures a properly colored (or rainbow) $K_{s,t}$.
A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of different colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(2020)] studied the existence of rainbow matchings in edge-colored graph G with average color degree at least 2k, and proved some sufficient conditions for a rainbow marching of size k in G. The sufficient conditions include that |V(G)|>=12k^2+4k, or G is a properly edge-colored graph with |V(G)|>=8k. In this paper, we show that every edge-colored graph G with |V(G)|>=4k-4 and average color degree at least 2k-1 contains a rainbow matching of size k. In addition, we also prove that every strongly edge-colored graph G with average degree at least 2k-1 contains a rainbow matching of size at least k. The bound is sharp for complete graphs.
Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $kgeq2$. This enables us to extend the $k$-colored partition function multiplicatively to a function on $k$-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.
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