Note on the group edge irregularity strength of graphs

Abstract We investigate the group edge irregularity strength (esg(G)) of graphs, i.e. the smallest value of s such that taking any Abelian group G of order s, there exists a function f : V ( G ) → G such that the sums of vertex labels at every edge are distinct. In this note we provide some the upper bounds on esg(G) as well as for edge irregularity strength es(G) and harmonious order har(G).

[1]  Florian Pfender,et al.  An iterative approach to graph irregularity strength , 2010, Discret. Appl. Math..

[2]  Bruce A. Reed,et al.  Vertex-Colouring Edge-Weightings , 2007, Comb..

[3]  Harri Haanp.  Sets in Zn with distinct sums of pairs , 2003 .

[4]  Florian Pfender,et al.  A New Upper Bound for the Irregularity Strength of Graphs , 2011, SIAM J. Discret. Math..

[5]  Martin Milanic,et al.  Group irregularity strength of connected graphs , 2015, J. Comb. Optim..

[6]  Joseph A. Gallian,et al.  Contemporary Abstract Algebra , 2021 .

[7]  Stanislav Jendrol',et al.  On irregular total labellings , 2007, Discret. Math..

[8]  N. J. A. Sloane,et al.  On Additive Bases and Harmonious Graphs , 1980, SIAM J. Algebraic Discret. Methods.

[9]  Ali Ahmad,et al.  On edge irregularity strength of graphs , 2014, Appl. Math. Comput..

[10]  Jitsuro Nagura.  On the interval containing at least one prime number , 1952 .

[11]  Alan M. Frieze,et al.  On graph irregularity strength , 2002, J. Graph Theory.

[12]  Patric R. J. Östergård,et al.  Sets in Z nwith distinct sums of pairs , 2004, Discret. Appl. Math..

[13]  Tchébichef,et al.  Mémoire sur les nombres premiers. , 1852 .

[14]  Piotr Majerski,et al.  On the Irregularity Strength of Dense Graphs , 2014, SIAM J. Discret. Math..

[15]  Jianguo Li,et al.  Neighbor sum distinguishing total chromatic number of planar graphs , 2018, Appl. Math. Comput..

[16]  Patric R. J. Östergård,et al.  Sets in Abelian groups with distinct sums of pairs , 2007 .

[18]  Joseph A. Gallian,et al.  Harmonious groups , 1991, J. Comb. Theory, Ser. A.

[19]  Till Nierhoff.  A Tight Bound on the Irregularity Strength of Graphs , 2000, SIAM J. Discret. Math..

[20]  Cun-Quan Zhang,et al.  The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture , 2016, J. Comb. Theory, Ser. B.

[21]  P. Erdos,et al.  On chromatic number of graphs and set-systems , 1966 .

[22]  J. Pintz,et al.  The Difference Between Consecutive Primes, II , 2001 .

[23]  Olivier Togni,et al.  Irregularity strength of trees , 1998, Discret. Math..

[24]  A. Rényi,et al.  ON THE REPRESENTATION OF THE NUMBERS 1,2,..., N BY MEANS OF DIFFERENCES, , 1965 .

[25]  Benny Sudakov,et al.  Embedding rainbow trees with applications to graph labelling and decomposition , 2018, Journal of the European Mathematical Society.

[26]  Bruce A. Reed,et al.  Degree constrained subgraphs , 2005, Discret. Appl. Math..

[27]  A. Thomason,et al.  Edge weights and vertex colours , 2004 .

[28]  Andrzej Zak.  Harmonious order of graphs , 2009, Discret. Math..

[29]  Eberhard Triesch,et al.  Irregular Assignments of Trees and Forests , 1990, SIAM J. Discret. Math..