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تسجيل مستخدم جديدRainbow triangles in edge-colored complete graphs
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Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $delta^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $delta^c(G)geq frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $delta^c(G)geq frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k$ rainbow triangles, which can be seen as a generalization of their result. Li showed that for an edge-colored graph $G$ of order $n$, if $delta^c(G)geq frac{n+1}{2}$, then $G$ contains a rainbow triangle. We show that if $G$ is complete and $delta^c(G)geq frac{n}{2}$, then $G$ contains a rainbow triangle and the bound is sharp. Hu et al. showed that for an edge-colored graph $G$ of order $ngeq 20$, if $delta^c(G)geq frac{n+2}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. We show that if $G$ is complete with order $ngeq 8$ and $delta^c(G)geq frac{n+1}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. Moreover, we improve the result of Hu et al. from $ngeq 20$ to $ngeq 7$, the best possible.
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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 t
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Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis ensures a rain
bow $ell$-cycle $C_{ell}$ whenever $n ge 432 ell$. This result is sharp for all odd integers $ell geq 3$, and extends earlier work of the authors for when $ell$ is even.
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $delta^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbo
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