A generalization of properly colored paths and cycles in edge-colored graphs
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On the existence of cycles with restrictions in the color transitions in edge-colored complete graphs
Consider the following edge-coloring of a graph G. Let H be a graph possibly with loops, an H-coloring of a graph G is defined as a function $$c : E(G) \rightarrow V(H).$$ c : E ( G ) → V ( H ) . We will say that G is an H-colored graph whenever we are taking a fixed H-coloring of G. A cycle $$(x_0,x_1,\ldots ,x_n,x_0),$$ ( x 0 , x 1 , … , x n , x 0 ) , in an H-colored graph, is an H-cycle if and only if $$(c(x_0x_1),c(x_1x_2),\ldots , c(x_nx_0),$$ ( c ( x 0 x 1 ) , c ( x 1 x 2 ) , … , c ( x n x 0 ) , $$c(x_0x_1))$$ c ( x 0 x 1 ) ) is a walk in H. Notice that the graph H determines what color transitions are allowed in a cycle in order to be an H-cycle, in particular, when H is a complete graph without loops, every H-cycle is a properly colored cycle. In this paper, we give conditions on an H-colored complete graph G, with local restrictions, implying that every vertex of G is contained in an H-cycle of length at least 5. As a consequence, we obtain a previous result about properly colored cycles. Finally, we show an infinite family of H-colored complete graphs fulfilling the conditions of the main theorem, where the graph H is not a complete k-partite graph for any k in $${\mathbb {N}}.$$ N .
On the existence of cycles with restrictions in the color transitions in edge-colored complete graphs
Consider the following edge-coloring of a graph G. Let H be a graph possibly with loops, an H-coloring of a graph G is defined as a function $$c : E(G) \rightarrow V(H).$$ c : E ( G ) → V ( H ) . We will say that G is an H-colored graph whenever we are taking a fixed H-coloring of G. A cycle $$(x_0,x_1,\ldots ,x_n,x_0),$$ ( x 0 , x 1 , … , x n , x 0 ) , in an H-colored graph, is an H-cycle if and only if $$(c(x_0x_1),c(x_1x_2),\ldots , c(x_nx_0),$$ ( c ( x 0 x 1 ) , c ( x 1 x 2 ) , … , c ( x n x 0 ) , $$c(x_0x_1))$$ c ( x 0 x 1 ) ) is a walk in H. Notice that the graph H determines what color transitions are allowed in a cycle in order to be an H-cycle, in particular, when H is a complete graph without loops, every H-cycle is a properly colored cycle. In this paper, we give conditions on an H-colored complete graph G, with local restrictions, implying that every vertex of G is contained in an H-cycle of length at least 5. As a consequence, we obtain a previous result about properly colored cycles. Finally, we show an infinite family of H-colored complete graphs fulfilling the conditions of the main theorem, where the graph H is not a complete k-partite graph for any k in $${\mathbb {N}}.$$ N .
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