Determining the total colouring number is NP-hard
Discrete Mathematics
Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
Total colouring regular bipartite graphs is NP-hard
Proceedings of the first Malta conference on Graphs and combinatorics
The $L(2,1)$-Labeling Problem on Graphs
SIAM Journal on Discrete Mathematics
Fixed parameter complexity of λ-labelings
Discrete Applied Mathematics - special issue on the 25th international workshop on graph theoretic concepts in computer science (WG'99)
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
k-L(2,1)-labelling for planar graphs is NP-complete for k≥4
Discrete Applied Mathematics
| Hi-index | 0.04 |
A (p,1)-total labelling of a graph G=(V,E) is a total colouring 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 @l"p(G). The case p=1 corresponds to the usual notion of total colouring, which is NP-hard to compute even for cubic bipartite graphs [C.J. McDiarmid, A. Sanchez-Arroyo, Total colouring regular bipartite graphs is NP-hard, Discrete Math. 124 (1994), 155-162]. In this paper we assume p=2. It is easy to show that @l"p(G)=@D+p-1, where @D is the maximum degree of G. Moreover, when G is bipartite, @D+p is an upper bound for @l"p(G), leaving only two possible values. In this paper, we completely settle the computational complexity of deciding whether @l"p(G) is equal to @D+p-1 or to @D+p when G is bipartite. This is trivial when @D@?p, polynomial when @D=3 and p=2, and NP-complete in the remaining cases.