H-Paths and H-Cycles in H-Coloured Digraphs

Abstract

A digraph D is a strongly transitive digraph if for any vertices ${u, v, w \in V(D)}$ (possibly u = w) such that ${\{(u, v), (v, w)\} \subseteq A(D)}$ implies ${(u, w) \in A(D)}$ (when u = w and ${\{(u, v), (v, w)\} \subseteq A(D)}$ then ${\{(u, u), (v, v)\} \subseteq A(D)}$). Let H be a strongly transitive digraph (possibly with loops) and D a digraph(that contains neither loops nor multiple arcs). A digraph D is said to be H-coloured if the arcs of D are coloured with the vertices of H. Will be denoted by c(x, y) the color of the arc ${(x, y)\in A(D)}$. A directed walk(directed path) ${C=(z_{0}, z_{1}, \ldots, z_{t})}$ in D will be called an H-walk(path) in D if ${(c(z_{0}, z_{1}), c(z_{1}, z_{2}), \ldots, c(z_{t-1}, z_{t}))}$ is a directed walk in H. Let D ₁ and D ₂ be spanning subdigraphs of D. A succession [u, v, w, u] is a (D ₁, D, D ₂) H-subdivision of C ₃, if there exist: T ₁ an H-path from u to v contained in D ₁, T an H-path from v to w in D, T ₂ an H-path from w to u contained in D ₂; and these path satisfies: (1) (c(final arc of T ₁), c(initial arc of T)) ${\notin A(H)}$, (c(final arc of T), c(initial arc of T ₂)) ${\notin A(H)}$ and (c(final arc of T ₂), c(initial arc of T ₁)) ${\notin A(H)}$; (2)${T_{1} \bigcup T\bigcup T_{2}}$ is a cycle in D. A succession [u, v, w, x ] is a (D ₁, D, D ₂) H-subdivision of P ₃, if there exist: T ₁ an H-path from u to v contained in D ₁, T an H-path from v to w in D, T ₂ an H-path from w to x contained in D ₂; such that (1) (c(final arc of T ₁), c(initial arc of T)) ${\notin A(H)}$ and (c(final arc of T), c(initial arc of T ₂)) ${\notin A(H)}$; (2) ${T_{1} \bigcup T\bigcup T_{2}}$ is a path in D. Let H be a strongly transitive digraph and D an H-coloured digraph. Let D ₁ and D ₂ be spanning subdigraphs of D. Will be said that P = {D ₁, D ₂} is an H-separation of D if: (1)${A(D_{1}) \bigcap A(D_{2})= \emptyset, A(D_{1}) \bigcup A(D_{2}) = A(D)}$; (2) every H-path of D is contained in D ₁ or it is contained in D ₂. In this paper will be proved that: if H is a strongly transitive digraph and D is an H-coloured digraph, P = {D ₁, D ₂} an H-separation of D such that: (1) every cycle of D that is contained in D _i is an H-cycle for ${i\in \{1, 2\}}$; (2) D does not contain a (D ₁, D, D ₂) H-subdivision of C ₃; (3) if (u, z, w, x ₀) is a (D ₁, D, D ₂) H-subdivision of P ₃ then there exists some H-path between u and x ₀. Then D has an H-kernel by paths.