I. Rosyida, W. Widodo, D. Indriati
Let G(V,E) be a graph. A function f from V (G) ∪E(G) to the set {1, 2, ⋯ k} is said to be a totally irregular total k-labeling of G if the weights of any two different vertices x and y in V (G) satisfy wf (x) ≠wf (y) where the weight wf (x) is the sum of label of x and labels of all edges incident to x, and the weights of any two different edges ux and vy in E(G) satisfy wf (ux) ≠wf (vy) where the weight wf (ux) is obtained from the sum of: label of x, label of u and label of edge ux. The total irregularity strength of the graph G, denoted by ts(G), is the minimum number k for which G has a totally irregular total k-labeling. In this paper, we focus on a caterpillar graph with two leaves on each internal vertex T2n+p,q, where n is the number of leaves on each end vertex of the central path, p is the number of leaves connected to internal vertices, q is the number of vertices of the central path, p ≤ 4 and q ≤ 4. Firstly, we do some experiments for constructing a formula for totally irregular total k-labeling of the caterpillar graph T2n+p,q. Secondly, we determine the minimum number k which is ts of the caterpillar graph T2n+p,q. We obtain that ts(T2(q-1)+p,q) = ⌈2(q-1)+p+1 2 m ⌉and ts(T2n+p,q) = ⌈2n+p+1 2 ⌉. © Published under licence by IOP Publishing Ltd.
Department of Mathematics, Universitas Negeri Semarang, Indonesia; Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia; Department of Mathematics, Universitas Sebelas Maret, Surakarta, Indonesia
