## Alternating Hamiltonian cycles.(English)Zbl 0325.05114

For natural numbers $$n$$ and $$d$$, let $$K_n(\Delta_c \leq d)$$ denote a complete graph of order $$n$$ whose edges are colored so that no vertex belongs to more than $$d$$ edges of the same color, and where $$\Delta_c$$ is the maximal degree in the subgraph formed by the edges of color $$c$$. D. E. Daykin proved that if $$d=2$$ and $$n \geq 6$$, then every such graph contains an alternating hamiltonian cycle (i.e. a spanning cycle whose adjacent edges have different colors). The authors have extended this as follows. Theorem: If $$69d <n$$, then every $$G=K_n(\Delta_c \leq d)$$ contains an alternating hamiltonian cycle. In fact, it is stated that if $$69d<n$$, then every $$G=K_n(\Delta_c \leq d)$$ contains alternating cycles of length $$\ell$$ for every $$\ell$$, $$3 \leq \ell \leq n$$. An analogous result is obtained as follows. Let $$\chi_v$$ denote the number of colors appearing among the edges containing the vertex $$v$$, and let $$K_n(\chi_v \geq \lambda)$$ denote a complete graph of order $$n$$ whose edges are colored so that each vertex is on at least $$\lambda$$ edges of different color. Theorem: Every $$K_n(\chi_v \geq (7/8)n)$$ contains an alternating hamiltonian cycle. Several related results and conjectures are also presented.
Reviewer: S.F.Kapoor

### MSC:

 05C35 Extremal problems in graph theory 05C15 Coloring of graphs and hypergraphs
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### References:

 [1] D. E. Daykin,Graphs with cycles having adjacent lines different colours, to appear. · Zbl 0287.05105 [2] P. Erdös and T. Gallai,On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar.10, (1959), 337–356. · Zbl 0090.39401
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