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arXiv 1301.0450math.COcs.DM
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边着色完全图中的长恰当着色圈

Long properly colored cycles in edge colored complete graphs

Guanghui Wang, Tao Wang, Guizhen Liu

备注：
8 pages
发表信息：
Discrete Mathematics, 324 (2014) 56--61
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摘要：

设$K_{n}^{c}$表示一个完全图，它的$n$顶点的边可以任意着色.令$Δ^{\mathrm{mon}} (K_{n}^{c})$表示与$K_{n}^{c}$的顶点关联的相同颜色的边的最大数目。$K_{n}^{c}$中正确着色的循环（路径）是相邻边具有不同颜色的循环（路径）。B. Bollobás和P. Erdös（1976）提出了如下猜想：如果$Δ^{\mathrm{mon}} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$，则$K_{n}^{c}$包含一个适当着色的哈密顿圈。Li，Wang和Zhou证明了，如果$Δ^{\mathrm{mon}} (K_{n}^{c})< \lfloor \frac{n}{2} \rfloor$，则$K_{n}^{c}$包含一个长度至少为$\lceil \frac{n+2}{3}\rceil+1$的适当着色圈.本文改进了$\lceil \frac{n}{2}\rceil + 2$的界。

英文摘要：

Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $Δ^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A properly colored cycle (path) in $K_{n}^{c}$ is a cycle (path) in which adjacent edges have distinct colors. B. Bollobás and P. Erdös (1976) proposed the following conjecture: if $Δ^{\mathrm{mon}} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if $Δ^{\mathrm{mon}} (K_{n}^{c})< \lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored cycle of length at least $\lceil \frac{n+2}{3}\rceil+1$. In this paper, we improve the bound to $\lceil \frac{n}{2}\rceil + 2$.

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