(p, 1)-Total labelling of graphs

A (p, 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) a vertex and its incident edge receive integers that differ by at least p in absolute value. The span of a (p, 1)-total labelling is the maximum difference between two labels. The minimum span of a (p, 1)-total labelling of G is called the (p, 1)-total number and denoted by T p (G). We provide lower and upper bounds for the (p, 1)-total number. In particular, generalizing the Total Colouring Conjecture, we conjecture that T p +2p-1 and give some evidences to support it. Finally, we determine the exact value of T p (Kn), except for even n in the interval [p + 5, 6p2 - 10p + 4] for which we show that T p (Kn) {n + 2p - 3, n + 2p - 2}.

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Frédéric Havet, Min-Li Yu
• Keywords: Adjacent Vertices, Distinct Integers, DM 2008, Labelling