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Register a new userPartitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition
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The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $ngeq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$
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We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-coloured random graph. More precisely, we show that for $pgg n^{-1/6}{(ln n)}^{1/6}$, in any $3$-edge-colouring of the random graph $G(n,p)$ we can find three monochromatic trees such that their union covers all vertices. This improves, for three colours, a result of Bucic, Korandi and Sudakov.
In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $delta^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $vin V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $ngeq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $ell$ for every $ell$ with $3 le ell le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $ngeq 3$ vertices with $delta^c(G)geq frac{n+1}{2}$ is properly vertex-pancyclic. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that $(G,c)$ does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.
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.
We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles. We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.
It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Delta^{mon}(G)leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker results for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain several partial solutions to a conjecture on disjoint cycles in directed graphs due to Bermond and Thomassen.
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