Maximal cliques in the Paley graph of square order

Baker, Rodney C.; Ebert, G. L.; Hemmeter, J.; Woldar, Andrew J.

doi:10.1016/s0378-3758(96)00006-7

Cited by 19 publications

(36 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let F (1) i,i be the set of frames containing both B i and B i and F (2) i,i be the set of frames containing both ϕ 2 (B i ) and ϕ 2 (B i ). There exists an injection f i,i from F (1) i,i to F (2) i,i . We use the rule (S2) for Stage 2 discharging.…”

Section: Densitymentioning

confidence: 99%

“…In particular, simultaneously bounding the independence number and clique number of circulant graphs has shown lower bounds on Ramsey numbers [30]. So far, these parameters have been studied for circulant graphs G(n, S) when limited to special classes of sets S, whether algebraically defined [1,4,6,12,20,28,40] or with S finite and n varying [3,7,23].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the independence ratio of distance graphs

Carraher¹,

Galvin²,

Hartke³

et al. 2016

Discrete Mathematics

View full text Add to dashboard Cite

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph G is the maximum density of an independent set in G. Lih, Liu, and Zhu [31] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs.We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.

show abstract

Section: Densitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the independence ratio of distance graphs

Carraher¹,

Galvin²,

Hartke³

et al. 2016

Discrete Mathematics

View full text Add to dashboard Cite

show abstract

“…An important step to proving these Cayley complements are r-primitive is to compute the clique number. Computing the clique number or independence number of a Cayley graph is very difficult, as many papers study this question [12,16], including in the special cases of circulant graphs [2,5,15,28] and Paley graphs [1,3,4,7]. Our enumerative approach to Theorem 2 and discharging approach to Theorem 3 provide a new perspective on computing these values.…”

Section: Algebraic Constructionsmentioning

confidence: 99%

“…But, there are 3t − 3 elements between B 0 and B k 1 , but at least 3t − 2 elements between B 0 and B t . Therefore, B 1 = B k 1 and there are t − 1 3-blocks between B 0 and B 1 , so 1…”

Section: Claim 12mentioning

confidence: 99%

Uniquely $K_r$-Saturated Graphs

Hartke

Stolee

2012

Electron. J. Combin.

View full text Add to dashboard Cite

A graph G is uniquely K r -saturated if it contains no clique with r vertices and if for all edges e in the complement, G + e has a unique clique with r vertices. Previously, few examples of uniquely K r -saturated graphs were known, and little was known about their properties. We search for these graphs by adapting orbital branching, a technique originally developed for symmetric integer linear programs. We find several new uniquely K r -saturated graphs with 4 ≤ r ≤ 7, as well as two new infinite families based on Cayley graphs for Z n with a small number of generators. IntroductionA graph G is uniquely H-saturated if there is no subgraph of G isomorphic to H, and for all edges e in the complement of G there is a unique subgraph in G + e isomorphic to H 4 . Uniquely Hsaturated graphs were introduced by Cooper, Lenz, LeSaulnier, Wenger, and West [9] where they classified uniquely C k -saturated graphs for k ∈ {3, 4}; in each case there is a finite number of graphs. Wenger [26,27] classified the uniquely C 5 -saturated graphs and proved that there do not exist any uniquely C k -saturated graphs for k ∈ {6, 7, 8}.In this paper, we focus on the case where H = K r , the complete graph of order r. Usually K r is the first graph considered for extremal and saturation problems. However, we find that classifying all uniquely K r -saturated graphs is far from trivial, even in the case that r = 4.Previously, few examples of uniquely K r -saturated graphs were known, and little was known about their properties. We adapt the computational technique of orbital branching into the graph theory setting to search for uniquely K r -saturated graphs. Orbital branching was originally introduced by Ostrowski, Linderoth, Rossi, and Smriglio [20] to solve symmetric integer programs. We further extend the technique to use augmentations which are customized to this problem. By executing this search, we found several new uniquely K r -saturated graphs for r ∈ {4, 5, 6, 7} and we provide constructions of these graphs to understand their structure. One of the graphs we discovered is a Cayley graph, which led us to design a search for Cayley graphs which are uniquely K r -saturated. Motivated by these search results, we construct two new infinite families of uniquely K r -saturated Cayley graphs.Erdős, Hajnal, and Moon [10] studied the minimum number of edges in a K r -saturated graph. They proved that the only extremal examples are the graphs formed by adding r − 2 dominating vertices to an independent set; these graphs are also uniquely K r -saturated. However, if G is uniquely K r -saturated and has a dominating vertex, then deleting that vertex results in a uniquely K r−1 -saturated graph. To avoid the issue of dominating vertices, we define a graph to be r-primitive if it is uniquely K r -saturated and has no dominating vertex. Understanding which r-primitive graphs exist is fundamental to characterizing uniquely K r -saturated graphs.Since K 3 ∼ = C 3 , the uniquely K 3 -saturated graphs were proven by Cooper et al.[9] to be stars an...

show abstract

“…In 1996, maximal cliques of order q+1 2 and q+3 2 for q ≡ 1(4) and q ≡ 3(4), respectively, were found [1] by Baker et al, but an exhaustive computer search done by them showed that these cliques are not the only cliques of such size. Moreover, there are no known maximal cliques whose size belongs to the gap from q+1 2 (from q+3 2 , respectively) to q.…”

Section: Introductionmentioning

confidence: 99%