Unitary Cayley graphs of Dedekind domain quotients

Sumun

2018

Discrete Mathematics

Self Cite

Let X Z/nZ denote the unitary Cayley graph of Z/nZ. We present results on the tightness of the known inequality γ(X Z/nZ ) ≤ γt(X Z/nZ ) ≤ g(n), where γ and γt denote the domination number and total domination number, respectively, and g is the arithmetic function known as Jacobsthal's function. In particular, we construct integers n with arbitrarily many distinct prime factors such that γ(X Z/nZ ) ≤ γt(X Z/nZ ) ≤ g(n) − 1. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.

Section: Introductionmentioning

confidence: 99%

Domination and upper domination of direct product graphs

Sumun

2018

Discrete Mathematics

Self Cite

“…This assertion does indeed hold, as the author proved in slightly greater generality in [12]. We mentioned before that the clique number of G Z/nZ is equal to the smallest prime factor of n; note that this follows as an easy corollary to the above formula (4).…”

Section: Unitary Cayley Graphs and Schemmel Totient Functionsmentioning

confidence: 71%

Enumerating cliques in direct product graphs

Journal of Combinatorics

2020

Self Cite

The unitary Cayley graph of Z/nZ, denoted G Z/nZ , is the graph with vertices 0, 1, . . . , n − 1 in which two vertices are adjacent if and only if their difference is relatively prime to n. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erdős and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order m in the unitary Cayley graph G Z/nZ . This formula involves an exciting class of arithmetic functions known as Schemmel totient functions, which we also briefly discuss. More generally, the proof yields a formula for the number of cliques of order m in a direct product of balanced complete multipartite graphs.

“…This means that it suffices to prove them in the case x = t n . Under this assumption, (18) and (19) become…”

Section: Independent Set Stabilitymentioning

confidence: 99%

Isoperimetry, stability, and irredundance in direct products

Alon

2020

Discrete Mathematics

Self Cite

The direct product of graphs G1, . . . , Gn is the graph with vertex set V (G1) × · · · × V (Gn) in which two vertices (g1, . . . , gn) and (g ′ 1 , . . . , g ′ n ) are adjacent if and only if gi is adjacent to g ′ i in Gi for all i. Building off of the recent work of Brakensiek, we prove an optimal vertex isoperimetric inequality for direct products of complete multipartite graphs. Applying this inequality, we derive a stability result for independent sets in direct products of balanced complete multipartite graphs, showing that every large independent set must be close to the maximal independent set determined by setting one of the coordinates to be constant. Armed with these isoperimetry and stability results, we prove that the upper irredundance number of a direct product of balanced complete multipartite graphs is equal to its independence number in all but at most 37 cases. This proves most of a conjecture of Burcroff that arose as a strengthening of a conjecture of the second author and Iyer. We also propose a further strengthening of Burcroff's conjecture.