Colorações backbone em grafos com galáxias backbone
A (proper) k-coloring of a graph G is a function φ: V (G) → {1, . . . , k} such that φ(u) ̸= φ(v), for all edge uv ∈ E(G). Given a graph G and a subgraph H ⊆ G, a q-backbone k-coloring of (G, H) is a k-coloring of G such that |φ(u)−φ(v)| ≥ q, for all edge uv ∈ E(H). The q-backbone chromatic number o...
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| Tipo de recurso: | tesis de maestría |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | Brasil |
| Institución: | Universidade Federal do Ceará (UFC) |
| Repositorio: | Repositório Institucional da Universidade Federal do Ceará (UFC) |
| Idioma: | portugués |
| OAI Identifier: | oai:repositorio.ufc.br:riufc/69637 |
| Acceso en línea: | http://www.repositorio.ufc.br/handle/riufc/69637 |
| Access Level: | acceso abierto |
| Palabra clave: | Coloração de grafos Coloração backbone circular Grafos planares Graph coloring Circular backbone coloring Planar graphs |
| Sumario: | A (proper) k-coloring of a graph G is a function φ: V (G) → {1, . . . , k} such that φ(u) ̸= φ(v), for all edge uv ∈ E(G). Given a graph G and a subgraph H ⊆ G, a q-backbone k-coloring of (G, H) is a k-coloring of G such that |φ(u)−φ(v)| ≥ q, for all edge uv ∈ E(H). The q-backbone chromatic number of (G, H), denoted by BBC q (G, H), is the smallest k ∈ Z such that there exists a q-backbone k-coloring of (G, H). A circular q-backbone k-coloring of (G, H) is a k-coloring of G such that q ≤ |φ(u) − φ(v)| ≤ k − q, for all edge uv ∈ E(H). The circular q-backbone chromatic number of (G, H), denoted by CBC q (G, H), is the smallest k ∈ Z such that there exists a circular q-backbone k-coloring of (G, H). In this dissertation, in addition to a brief presentation of the results related to Backbone Coloring, we present our contributions, among which we partially answer three problems proposed in (Havet, Frédéric et al., 2014): we show that if G is a planar graph with a spanning subgraph H, then CBC q (G, H) ≤ 2q + 2 when q ≥ 3 and H is a galaxy; CBC q (G, H) ≤ 2q when q ≥ 4 and H is a matching; and, CBC 3 (G, H) ≤ 7 when G does not have a pair of triangles with adjacent edges and H is a matching. Some of these results follow as a consequence of more general results we obtained about the parameter CBC q (G, H) for graph classes larger than the class of planar graphs. In addition, we show that it is possible to determine BBC q (G, H) and CBC q (G, H) in polynomial time when G has bounded treewidth graph and H is a matching of G. Finally, we present an error in the demonstration that BBC 2 (G, H) ≤ ∆(G) + 1, for any matching H in an arbitrary graph G (Miskuf, Jozef et al., 2010), and we present a demonstration for this result. |
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