Product irregularity strength of certain graphs

Abstract

Consider a simple graph G with no isolated edges and at most one isolated vertex. A labeling w : E(G) → {1, 2, . . . , m} is called product -irregular, if all product degrees pd G (v) = e v w(e) are distinct. The goal is to obtain a product -irregular labeling that minimizes the maximal label. This minimal value is called the product irregularity strength and denoted ps(G). We give the exact values of ps(G) for several families of graphs, as complete bipartite graphs K m,n , where 2 ≤ m ≤ n ≤ m+2 2, some families of forests, including complete d-ary trees, and other graphs with δ(G) = 1.

Product irregularity strength of graphs with small clique cover number

“…Anholcer in [2] considered product irregularity strength of complete bipartite graphs and forests. Anholcer proved that for two integers m and n such that 2 ≥ m ≥ n it holds ps(K m,n ) = 3 if and only if n ≥ m+2…”

“…If pd(a (3) ) = pd(c(1) ) then (n − 3, 1) = (1, m−1 , so n = 4 which is a contradiction. If pd(a (3) ) = pd(c(2) ) then (n − 3, 1) = (0, m − j) or (n − 3, 1) = (0, m − j + 1), so n = 3 which is a contradiction. If pd(a (3) ) = pd(c (3) ) then (n − 3, 1) = (1, 1), so n = 4 which is a contradiction.This finishes the proof.…”

“…. The main result in [2] is about product irregularity strength of almost all forests F such that (F ) = D for arbitrary integer D ≥ 3, n 2 = 0, n 0 ≤ 0 and n 2 = 0 of the forest F with all pendant edges removed, where n d denotes the number of vertices of degree d. Anholcer proved that in this case ps(F ) = n 1 .…”

“…If pd(a (1) ) = pd(c (3) ) then ( n 2 − 2, 1) = (1, 1), so n = 5 or n = 6 which is a contradiction. If pd(a(2) ) = pd(c (1) ) then (i − 1, 0) = (1, m−1 or (i − 2, 0) = (1, m−1 2). In both cases m = 1 which is a contradiction.5.…”

Graphs with multiplicative vertex-coloring 2-edge-weightings

“…Some authors distinguish all the vertices in a graph by their product colors (product irregularity strength of graphs, see Anholcer (2009Anholcer ( , 2014, Darda and Hujdurovic (2014)). The total version of this problem was also investigated, where all the edges and vertices are weighted with integers, and the color of a vertex is the sum of its weight and the weights of the incident edges (see Przybyło and Woźniak (2010), Skowronek-Kaziów (2008)).…”