New Examples of Graphs without Small Cycles and of Large Size (original) (raw)

New families of graphs without short cycles and large size

2010

We denote by ex(n; {C 3 , C 4 , . . . , C s }) or f s (n) the maximum number of edges in a graph of order n and girth at least s + 1. First we give a method to transform an n-vertex graph of girth g into a graph of girth at least g − 1 on fewer vertices. For an infinite sequence of values of n and s ∈ {4, 6, 10} the obtained graphs are denser than the known constructions of graphs of the same girth s + 1. We also give another different construction of dense graphs for an infinite sequence of values of n and s ∈ {7, 11}. These two methods improve the known lower bounds on f s (n) for s ∈ {4, 6, 7, 10, 11} which were obtained using different algorithms. Finally, to know how good are our results, we have proved that lim sup n→∞

Properties of Certain Families of 2k-Cycle-Free Graphs

Journal of Combinatorial Theory, Series B, 1994

Let v = v(G) and e = e(G) denote the order and size of a simple graph G, respectively. Let G = {G i } i≥1 be a family of simple graphs of magnitude r > 1 and constant λ > 0, i.e. e(G i) = (λ + o(1))v(G i) r , i → ∞. For any such family G whose members are bipartite and of girth at least 2k + 2, and every integer t, 2 ≤ t ≤ k − 1, we construct a family G t of graphs of same magnitude r, of constant greater than λ, and all of whose members contain each of the cycles C 4 , C 6 ,. .. , C 2t , but none of the cycles C 2t+2 ,. .. , C 2k. We also prove that for every family of 2k-cycle free extremal graphs (i.e. graphs having the greatest size among all 2k-cycle free graphs of the same order), all but finitely many such graphs must be either non-bipartite or have girth at most 2k − 2. In particular, we show that the best known lower bound on the size of 2k-cycle free extremal graphs for k = 3, 5, namely (2 − k+1 k + o(1))v k+1 k , can be improved to ((k − 1) • k − k+1 k + o(1))v k+1 k .

VERTEX-TRANSITIVE GRAPHS OF ORDER 2p

Annals of the New York Academy of Sciences, 1979

In the past ten years there has been a considerable amount of activity in the area of circulant graphs and digraphs. Most of this has consisted of investigation of basic properties of circulants along with some applications. We shall now summarize some of this activity.

A Construction of Small (q−1)(q-1)(q−1)-Regular Graphs of Girth 8

The Electronic Journal of Combinatorics, 2015

In this note we construct a new infinite family of (q−1)(q-1)(q−1)-regular graphs of girth 8 and order 2q(q−1)22q(q-1)^22q(q−1)2 for all prime powers qgeq16q\geq 16qgeq16, which are the smallest known so far whenever q−1q-1q−1 is not a prime power or a prime power plus one itself.

Edge-girth-regular graphs arising from biaffine planes and Suzuki groups

Discrete Mathematics

An edge-girth-regular graph egr(v, k, g, λ), is a k-regular graph of order v, girth g and with the property that each of its edges is contained in exactly λ distinct g-cycles. An egr(v, k, g, λ) is called extremal for the triple (k, g, λ) if v is the smallest order of any egr(v, k, g, λ). In this paper, we introduce two families of edge-girth-regular graphs. The first one is a family of extremal egr(2q 2 , q, 6, (q − 1) 2 (q − 2)) for any prime power q ≥ 3 and, the second one is a family of egr(q(q 2 + 1), q, 5, λ) for λ ≥ q − 1 and q ≥ 8 an odd power of 2. In particular, if q = 8 we have that λ = q − 1. Finally, we construct an egr(32, 5, 5, 12) and we prove that it is extremal.

On locally projective graphs of girth 5

Journal of Algebraic Combinatorics, 1998

Let be a graph and G be a 2-arc transitive automorphism group of . For a vertex x ∈ let G(x) (x) denote the permutation group induced by the stabilizer G(x) of x in G on the set (x) of vertices adjacent to x in . Then is said to be a locally projective graph of type (n, q) if G(x) (x) contains PSL n (q) as a normal subgroup in its natural doubly transitive action. Suppose that is a locally projective graph of type (n, q), for some n ≥ 3, whose girth (that is, the length of a shortest cycle) is 5 and suppose that G(x) acts faithfully on (x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n = 4, q = 2, has 506 vertices and G ∼ = M 23 , or q = 4, PSL n (4) ≤ G(x) ≤ PGL n (4), and contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W (n) of which all other graphs for this n are quotients. The graph W (3) satisfies the conditions and has 2 20 vertices.

Some algebraic constructions of dense graphs of large girth and of large size

Let k ≥ 3 be a positive odd integer and q be a power of a prime. In this paper we give an explicit construction of a q-regular bipartite graph on v = 2q k vertices with girth g ≥ k + 5. The constructed graph is the incidence graph of a flag-transitive semiplane. For any positive integer t we also give an example of a q = 2 t-regular bipartite graph on v = 2q k+1 vertices with girth g ≥ k + 5 which is both vertex-transitive and edge-transitive .

Families of small regular graphs of girth 7

In this paper we obtain (q + 3)-regular graphs of girth 5 with fewer vertices than previously known ones for q = 13, 17, 19 and for any prime q ≥ 23 performing operations of reductions and amalgams on the Levi graph Bq of an elliptic semiplane of type C.

Decompositions of some classes of regular graphs and digraphs into cycles of length 4p

Australas. J Comb., 2021

In this paper, we prove the existence of a 4p-cycle decomposition of the graph Km ×Kn and a directed 4p-cycle decomposition of the symmetric digraph (Km◦Kn)∗, where ◦ and × denote the wreath product and tensor product of graphs, respectively, and p is an odd prime. It is proved that, for integers m ≥ 3 and n ≥ 3, the obvious necessary conditions for the existence of a 4p-cycle decomposition of Km ×Kn are sufficient, where p is an odd prime. Also, it is shown that the necessary conditions for the existence of a directed 4p-cycle decomposition of the symmetric digraph (Km ◦Kn)∗ are sufficient, where p is an odd prime. Recently, the same type of results are obtained for 2p; see [S. Ganesamurthy and P. Paulraja, Discrete Math. 341 (2018), 2197–2210]. ISSN: 2202-3518 c ©The author(s). Released under the CC BY 4.0 International License S. GANESAMURTHY ET AL. /AUSTRALAS. J. COMBIN. 79 (2) (2021), 215–233 216