Andrew J. Woldar

A new series of dense graphs of high girth

Let k ≥ 1 be an odd integer, t = b k+2 4 c, and q be a prime power. We construct a bipartite, q–regular, edge–transitive graph CD(k, q) of order v ≤ 2qk−t+1 and girth g ≥ k + 5. If e is the the number of edges of CD(k, q), then e = Ω(v 1 k−t+1 ). These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g ≥ 5, g 6= 11, 12. For g ≥ 24, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for 5 ≤ g ≤ 23, g 6= 11, 12, it improves on or ties existing bounds.

Polarities and 2 k -cycle-free graphs

Abstract Let C2k be the cycle on 2k vertices, and let ex(v, C2k) denote the greatest number of edges in a simple graph on v vertices which contains no subgraph isomorphic to C2k. In this paper we discuss a method which allows one to sometimes improve numerical constants in lower bounds for ex(v, C2k). The method utilizes polarities in certain rank two geometries. It is applied to refute some conjectures about the values of ex(v, C2k), and to construct some new examples of graphs having certain restrictions on the lengths of their cycles. In particular, we construct an infinite family {Gi} of C6-free graphs with |E(G i )| ∼ 1 2 |V(G i )| 4 3 , i → ∞ , which improves the constant in the previous best lower bound on ex(v, C6) from 2 3 4 3 ≈ 0.462 to 1 2 .

A characterization of the components of the graphs D ( k,q )

Abstract We study the graphs D(k,q) of [4] with particular emphasis on their connected components when q is odd. In [6] the authors proved that these components (most often) provide the best-known asymptotic lower bound for the greatest number of edges in graphs of their order and girth. It was further shown in [6] that D(k,q) has at least qt−1 components, where t = ⌊(k + 2)/4⌋. In this paper we prove that the value qt−1 is precise and that the numerical invariant introduced in [6] completely characterizes the components of D(k,q). Some general results regarding the relationship between D(l,q) and D(k,q) (l

Graphs of prescribed girth and bi-degree

Abstract We say that a bipartite graph Γ ( V 1 ∪ V 2 , E ) has bi-degree r , s if every vertex from V 1 has degree r and every vertex from V 2 has degree s . Γ is called an ( r , s , t )-graph if, additionally, the girth of Γ is 2 t . For t > 3, very few examples of ( r , s , t )-graphs were previously known. In this paper we give a recursive construction of ( r , s , t )-graphs for all r , s , t ≥ 2, as well as an algebraic construction of such graphs for all r , s ≥ t ≥ 3.

New Lower Bounds on the Multicolor Ramsey Numbers rk(C4)

The multicolor Ramsey number rk(C4) is the smallest integer n for which any k-coloring of the edges of the complete graph Kn must produce a monochromatic 4-cycle. It was proved earlier that rk(C4)?k2?k+2 for k?1 being a prime power. In this note we establish rk(C4)?k2+2 for k being an odd prime power.

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

Let v = v(G) and e = e(G) denote the order and size of a simple graph G, respectively. Let G = {Gi}i?1, be a family of simple graphs of magnitude r > 1 and constant ? > 0, i.e., e(Gi) = (? + o(1))v(Gi)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 the same magnitude r, of constant greater than ?, and all of whose members contain each of the cycles C4, C6, ..., C2t, but none of the cycles C2t + 2, ..., C2k. 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.

New constructions of bipartite graphs on m, n vertices with many edges and without small cycles

Abstract For arbitrary odd prime power q and s ∈ (0, 1] such that qs is an integer, we construct a doubly-infinite series of (q5, q3 + s)-bipartite graphs which are biregular of degrees qs and q2 and of girth 8. These graphs have the greatest number of edges among all known (n, m)-bipartite graphs with the same amsymptotics of lognm, n → ∞. For s = 1 3 , our graphs provide an explicit counterexample to a conjecture of Erdős which states that an (n, m)-bipartite graph with m = O(n 2 3 ) and girth at least 8 has O(n) edges. This conjecture was recently disproved by de Caen and Szekely, who established the existence of a family of such graphs having n1 + 1 57 + o(1) edges. Our graphs have n1 + 1 15 edges, and so come closer to the best known upper bound of O(n1 + 1 9 ).

Maximal cliques in the Paley graph of square order

Copyright (c) 1996 Elsevier Science B.V. All rights reserved. Determining the clique number of the Paley graph of order q, q≡1 (mod 4r a prime power, is a difficult problem. However, the work of Blokhuis implies that in the Paley graph of order q 2 , where q is any odd prime power, the clique number is in fact q. In this paper we construct maximal cliques of size (1r/(2r(q+1r or (1r/(2r(q+3r, accordingly as q≡1 (mod 4r or q≡3 (mod 4r, in the Paley graph of order q 2 . It is believed that these are the largest maximal cliques which are not maximum. We also briefly discuss maximal cliques in some graphs naturally associated with the interior and exterior points of a conic in PG(2,qr for odd prime powers q.

Extremal properties of regular and affine generalized m -gons as tactical configurations

The purpose of this paper is to derive bounds on the sizes of tactical configurations of large girth which provide analogues to the well-known bounds on the sizes of graphs of large girth. Let exα(v,g) denote the greatest number of edges in a tactical configuration of order v, bidegree a, aα and girth at least g. We establish the upper bound exα (v,g) = 0(v1+1/τ), where τ = 1/4(α + 1)g -1 for g = 0(mod 4) and τ = 1/4(α + 1)g + ½(α - 3) for g ≡ = 2(mod 4). We further demonstrate this bound to be sharp for the regular and affine generalized m-gons but not for the nonregular generalized m-gons. Finally, we derive lower bounds on exα(v, g) via explicit group theoretic constructions.

Examples of computer experimentation in algebraic combinatorics

We introduce certain paradigms for procuring computer-free explanations from data acquired via computer algebra experimentation. Our established context is the field of algebraic combinatorics, with special focus on coherent configurations and association schemes. All results presented here were obtained by the authors with the aid of computer algebra systems, especially COCO and GAP. A number of examples are explored, in particular of objects on 28, 50, 63, and 210 points. In a few cases, initial experimental data pointed to appropriate theoretical generalizations that yielded an infinite class of related combinatorial structures. Special attention is paid to algebraic automorphisms (of a coherent algebra), a fairly new concept that has already proved to have far-reaching consequences. Finally, we focus on the Doyle-Holt graph on 27 vertices, and some of its related structures.