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Backbone coloring of graphs with galaxy backbones
被引:0
|作者:
Araujo, C. S.
[1
]
Araujo, J.
[1
,2
]
Silva, A.
[1
]
Cezar, A. A.
[1
]
机构:
[1] Univ Fed Ceara, Dept Matemat, ParGO, Fortaleza, Brazil
[2] Univ Montpellier, LIRMM, Montpellier, France
关键词:
Graph coloring;
Backbone coloring;
Brooks' type theorem;
TREE BACKBONES;
D O I:
10.1016/j.dam.2022.06.024
中图分类号:
O29 [应用数学];
学科分类号:
070104 ;
摘要:
A (proper) k-coloring of a graph G = (V, E) is a function c : V (G) -> {1, ... , k} such that c(u) not equal c(v), for every uv is an element of E(G). Given a graph G and a spanning subgraph H of G, a (circular) q-backbone k-coloring of (G, H) is a k-coloring c of G such that q <= vertical bar c(u) - c(v)vertical bar (q <= vertical bar c(u) - c(v)vertical bar <= k-q), for every edge uv is an element of E(H). The (circular) q-backbone chromatic number of (G, H), denoted by BBCq(G, H) (CBCq(G, H)), is the minimum integer k for which there exists a (circular) q-backbone k-coloring of (G, H). In this work, we (partially) answer three questions posed by Havet et al. (2014), namely, we prove that if G is a planar graph, H is a spanning subgraph of G and q is a positive integer, then: CBCq(G, H) <= 2q + 2 when q >= 3 and H is a galaxy; CBCq(G, H) <= 2q when q >= 4 and H is a matching; and CBC3(G, H) <= 7 when H is a matching and G has no triangles sharing an edge. In addition, we present a polynomial-time algorithm to determine both parameters for any pair (G, H), whenever G has bounded treewidth. Finally, we show how to fix a mistake in a proof that BBC2(G, M) <= Delta(G) + 1, for any matching M of an arbitrary graph G (Miskuf et al., 2010). (c) 2022 Elsevier B.V. All rights reserved.
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页码:2 / 13
页数:12
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