The irregularity strength of tP3

AbstractLet (a1,…,at, b1,…,bt) be the sequence of distinct positive integers such that ai+ bi are distinct for i = 1,…,t, and different from aj and bj, 1 ⩽ j ⩽ t. Denote by s(t) the minimum of the largest element of these sequences for fixed t. In this note we prove s(t) ⩾ ⌈(15t − 1)/7⌉ and exhibit infinitely many sequences attaining equality. We also show s(t) ⩽ ⌈(15t − 1)/7⌉ + 1 for every t. As a corollary we obtain that the irregularity strength of the graph G = tP3, the disjoint union of t paths of length 3, is about 5n/7, where n = 3t is the order of G

An iterative approach to graph irregularity strength

AbstractAn assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we take an iterative approach to calculating the irregularity strength of a graph. In particular, we develop a new algorithm that determines the exact value s(T) for trees T in which every two vertices of degree not equal to two are at distance at least eight

On H-irregularity Strengths of G-amalgamation of Graphs

A simple graph G=(V(G),E(G)) admits an H-covering if every edge in E(G) belongs at least to one subgraph of G isomorphic to a given graph H. Then the graph G admitting H-covering admits an H-irregular total k-labeling f: V(G) U E(G) \to {1, 2, ..., k} if for every two different subgraphs H\u27 and H\u27\u27 isomorphic to H there is $wt_{f}(H\u27) \neq wt_{f}(H\u27\u27)$, where $wt_{f}(H)= \sum \limits_{v\in V(H)} f(v) + \sum \limits_{e \in E(H)} f(e)$ is the associated H-weight. The minimum k for which the graph G has an H-irregular total k-labeling is called the total H-irregularity strength of the graph G.In this paper, we obtain the precise value of the total H-irregularity strength of G-amalgamation of graphs