On the irregularity strength of trees

Abstract

For any graph G, let n i be the number of vertices of degree i, and ðGÞ ¼ max i j f n i þÁÁÁþn j þiÀ1 j g. This is a general lower bound on the irregularity strength of graph G. All known facts suggest that for connected graphs, this is the actual irregularity strength up to an additive constant. In fact, this was conjectured to be the truth for regular graphs and for trees. Here we find an infinite sequence of trees with ðT Þ ¼ n 1 but strength converging to 11À ffiffi 5 p 8 n 1 . ß

On a conjecture about edge irregular total labelings

“…This parameter has attracted much attention [1,2,5,6,8,13] and motivated by this research, Bača et al [4] recently defined an edge irregular total k-labeling of a graph G = (V, E) to be a labeling of the vertices and edges of G f : V ∪ E → {1, 2, . .…”

Irregularity strength of dense graphs

“…For many results which were not mentioned in the survey and appeared since it was published, see [1], [11], [16], [12], [10], [2], [17], [21], [20], [26], [3], [18], [23], [25], [15], [5], [4].…”

Color-Induced Graph Colorings

“…For example, the papers [38,41,42] deal with the irregularity strength of regular graphs and [6,13] concern trees. The papers [25,40] discuss the irregularity strength of dense graphs (those graphs of order n and size m for which m=n is large).…”