Complexity of (p, 1)-total labelling
نویسندگان
چکیده
A (p, 1)-total labelling of a graph G = (V,E) is a total coloring L from V ∪ E into {0, . . . , l} such that |L(v) − L(e)| ≥ p whenever an edge e is incident to a vertex v. The minimum l for which G admits a (p, 1)-total labelling is denoted by λp(G). The case p = 1 corresponds to the usual notion of total colouring, which is NP-hard to compute even for cubic bipartite graphs [6]. In this paper we assume p ≥ 2. It is easy to show that λp(G) ≥ ∆ + p − 1, where ∆ is the maximum degree of G. Moreover, when G is bipartite, ∆+p is an upper bound for λp(G), leaving only two possible values. In this paper, we completely settle the computational complexity of deciding whether λp(G) is equal to ∆ + p − 1 or to ∆ + p when G is bipartite. This is trivial when ∆ ≤ p, polynomial when ∆ = 3 and p = 2, and NP-complete in the remaining cases.
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عنوان ژورنال:
- Discrete Applied Mathematics
دوره 157 شماره
صفحات -
تاریخ انتشار 2009