Upper bound for the span of (s,1)-total labelling of graphs - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Reports Year : 2003

Upper bound for the span of (s,1)-total labelling of graphs


An (s,1)-total labelling of a graph G is an assignment of integers to V(G)E(G) such that: (i) any two adjacent vertices of G receive distinct integers, (ii) any two adjacent edges of G receive distinct integers, and (iii) incident vertex and edge receive integers that differ by at least s in absolute value. The span of a (s,1)-total labelling is the maximum difference between two labels. The minimum span of a (s,1)-total labelling of G is denoted by _s^T(G). In , it is conjectured that _s^T+ 2s -1, where is the maximum degree of a vertex in a graph. This is an extension of the Total Colouring Conjecture. It is also shown that _s^T2+ s -1 and _2^T(G)6 if (G)3 and _2^T(G)8 if (G)4. In this paper, we prove that _s^T2-(+2) + s -1 +2(16s-10). The proof is an induction based on the maximal cut of a graph. We use the same technique to improve a little bit this result in the case of (2,1)-total labelling. We prove that if (G) 3, then _2^T(G)2(G) and that if (G)5 is odd then _2^T(G)2(G)-1.


Other [cs.OH]
Fichier principal
Vignette du fichier
RR-4816.pdf (194.29 Ko) Télécharger le fichier

Dates and versions

inria-00071770 , version 1 (23-05-2006)


  • HAL Id : inria-00071770 , version 1


Frédéric Havet. Upper bound for the span of (s,1)-total labelling of graphs. RR-4816, INRIA. 2003. ⟨inria-00071770⟩
178 View
17964 Download


Gmail Facebook Twitter LinkedIn More