## Cycles of given color patterns.(English)Zbl 0839.05056

A 2-edge-colored complete graph $$K^c_n$$ is a complete graph $$K_n$$ with edges colored in colors 1 and 2. An $$(s, t)$$-cycle in $$K^c_n$$ is a cycle of length $$s+ t$$, in which $$s$$ consecutive edges are in one color and the remaining $$t$$ edges are in the other color. This paper gives a characterization for the existence of $$(s, t)$$-cycles in $$K^c_n$$ and studies all possible $$(s, t)$$-cycles of length 4 and shows that $$K^c_n$$ contains an $$(s, t)$$-Hamiltonian cycle unless it is isomorphic to a specified graph. This extends a result of A. Gyárfás.

### MSC:

 05C38 Paths and cycles 05C15 Coloring of graphs and hypergraphs 05C45 Eulerian and Hamiltonian graphs

### Keywords:

2-edge-colored complete graph; $$(s, t)$$-cycle; color
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