Abstract
Let \(K_n^c\) denote a complete graph on n vertices whose edges are colored in an arbitrary way. And let \(\Delta (K_n^c)\) denote the maximum number of edges of the same color incident with a vertex of \(K_n^c\). A properly colored cycle (path) in \(K_n^c\), that is, a cycle (path) in which adjacent edges have distinct colors is called an alternating cycle (path). Our note is inspired by the following conjecture by B. bollobás and P. Erdős(1976): If \(\Delta(K_n^c)<\lfloor n/2\rfloor\), then \(K_n^c\) contains an alternating Hamiltonian cycle. We prove that if \(\Delta (K_n^c)< \lfloor n/2\rfloor\), then \(K_n^c\) contains an alternating cycle with length at least \(\lceil \frac{n+2}{3}\rceil+1\).
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Li, H., Wang, G., Zhou, S. (2007). Long Alternating Cycles in Edge-Colored Complete Graphs. In: Preparata, F.P., Fang, Q. (eds) Frontiers in Algorithmics. FAW 2007. Lecture Notes in Computer Science, vol 4613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73814-5_29
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DOI: https://doi.org/10.1007/978-3-540-73814-5_29
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