(2, 1)-Total Labeling of a Class of Subcubic Graphs

A (p, 1)-total labeling 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 an edge incident receive integers that differ by at least p in absolute value. The span of a (p, 1)-total labeling is the maximum difference between two labels. The minimum of span of all possible (p, 1)-total labeling of G is called the (p, 1)-total number and denoted by . The well known Havet and Yu conjecture states that for any connected graph G with Δ(G)≤3 and G≠K4, . In this paper, we determine the (2, 1)-total number of a connected graph G with the property that Δ(G)=3 and the vertices of G are covered by a set of independent triangles of G. More precisely, we show that for such subcubic graphs G, . This result supports Havet and Yu conjecture.