Linear bounds on nowhere-zero group irregularity strength and nowhere-zero group sum chromatic number of graphs
Marcin Anholcer , Sylwia Cichacz , Jakub Przybyło
AbstractWe investigate the group irregularity strength, sg(G), of a graph, i.e., the least integer k such that taking any Abelian group of order k, there exists a function so that the sums of edge labels incident with every vertex are distinct. So far the best upper bound on sg(G) for a general graph G was exponential in where n is the order of G and c denotes the number of its components. In this note we prove that sg(G) is linear in n, namely not greater than 2n. In fact, we prove a stronger result, as we additionally forbid the identity element of a group to be an edge label or the sum of labels around a vertex. We consider also locally irregular labelings where we require only sums of adjacent vertices to be distinct. For the corresponding graph invariant we prove the general upper bound: (where col(G) is the coloring number of G) in the case when we do not use the identity element as an edge label, and a slightly worse one if we additionally forbid it as the sum of labels around a vertex. In the both cases we also provide a sharp upper bound for trees and a constant upper bound for the family of planar graphs.
|Journal series||Applied Mathematics and Computation, ISSN 0096-3003, e-ISSN 1873-5649, (N/A 100 pkt)|
|Publication size in sheets||0.5|
|Keywords in English||Irregularity strength, Sum chromatic number, Coloring number, Arboricity, Abelian group|
|Score||= 100.0, 04-03-2020, ArticleFromJournal|
|Publication indicators||= 0; : 2018 = 1.544; : 2018 = 3.092 (2) - 2018=2.429 (5)|
|Citation count*||1 (2020-06-25)|
* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perish system.