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Characterizing arc-colored digraphs with an Eulerian trail with restrictions in the color transitions | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 7، دوره 12، شماره 1، خرداد 2027، صفحه 77-91 اصل مقاله (469.8 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.29828.2178 | ||
| نویسندگان | ||
| Carlos Vilchis-Alfaro* ؛ Hortensia Galeana-Sánchez | ||
| Instituto de Matemáticas, Universidad Nacional Autónama de México, CDMX, México | ||
| چکیده | ||
| Let $H$ be a digraph possibly with loops, and $D$ a multidigraph without loops. An $H$-coloring of $D$ is a function $c: A(D) \rightarrow V(H)$. We say that $D$ is an $H$-colored multidigraph whenever we are taking a fixed $H$-coloring of $D$. A trail $W=(v_0,e_0,v_1,e_1,v_2,\ldots,v_{n-1},e_{n-1},$ $v_n)$ in $D$ is an $H$-trail if and only if $(c(e_i),c(e_{i+1}))$ is an arc in $H$, for each $i \in \{0,\ldots,n-2\}$. We say that an $H$-colored multidigraph is $H$-trail-connected if and only if there is an $H$-trail starting with arc $f_1$ and ending with arc $f_2$, for any pair of arcs $f_1$ and $f_2$ in $D$. Let $D$ be an $H$-colored multidigraph and $u$ a vertex of $D$, the auxiliary digraph $D_u$ is the digraph of allowed transition throughout $u$. In this paper we give the following characterization: Let $D$ be an $H$-colored multidigraph such that the underlying graph of $D_u$ is a disjoint union of complete bipartite graphs, for every $u \in V(D)$. Then $D$ has a Euler $H$-trail if and only if $D$ is $H$-trail-connected and, for every $u \in V(D)$, the underlying graph of $D_u$ has a perfect matching. As a consequence we obtain the well-known characterization of the 2-arc-colored multidigraphs containing properly colored Euler trail. Finally, we give an infinite family of digraphs $H$ such that for every multidigraph $D$ without isolated vertices, and every $H$-coloring of $D$, the underlying graph of $D_u$ is a disjoint union of complete bipartite graphs and, possibly, isolated vertices, for every $u \in V(D)$. | ||
| کلیدواژهها | ||
| Euler trails؛ arc-colored digraphs؛ forbidden transitions | ||
| مراجع | ||
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