Connectivity of graphs with given girth pair (original) (raw)

On the connectivity of certain graphs of high girth

Discrete Mathematics, 2004

Let q be a prime power and k ≥ 2 be an integer. In [2] and [3] it was determined that the number of components of certain graphs D(k, q) introduced in [1] is at least q t−1 where t = k+2 4. This implied 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. In [4], it was shown that the number of components is (exactly) q t−1 for q odd, but the method used there failed for q even. In this paper we prove that the number of components of D(k, q) for even q > 4 is again q t−1 where t = k+2 4. Our proof is independent of the parity of q as long as q > 4. Furthermore, we show that for q = 4 and k ≥ 4, the number of components is q t .

On the λ′-optimality in graphs with odd girth g and even girth h

2011

For a connected graph G, the restricted edge-connectivity λ ′ (G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in }, d(u) denoting the degree of a vertex u. The main result of this paper is that graphs with odd girth g and finite even girth h ≥ g + 3 of diameter at most h − 4 are λ ′ -optimal. As a consequence polarity graphs are shown to be λ ′ -optimal.

Nonexistence of certain edge-girth-regular graphs

arXiv (Cornell University), 2024

Edge-girth-regular graphs (abbreviated as egr graphs) are regular graphs in which every edge is contained in the same number of shortest cycles. We prove that there is no 3-regular egr graph with girth 7 such that every edge is on exactly 6 shortest cycles, and there is no 3-regular egr graph with girth 8 such that every edge is on exactly 14 shortest cycles. This was conjectured by Goedgebeur and Jooken [2]. A few other unresolved cases are settled as well.

Small bi-regular graphs of even girth

Discrete Mathematics, 2016

The number of vertices of a graph of diameter two and maximum degree d is at most d 2 + 1. This number is the Moore bound for diameter two. The order of largest Cayley graphs of diameter two and degree d is denoted by C(d, 2). The only known construction of Cayley graphs of diameter 2 valid for all degrees d gives C(d, 2) > 1 4 d 2 +d. However, there is a construction yielding Cayley graphs of diameter 2, degree d and order d 2 − O(d 3 2) for an infinite set of degrees d of a special type [1]. We present a construction giving C(d, 2) ≥ 1 2 d 2 − k for d even and of order C(d, 2) 1 2 (d 2 + d) − k for d odd, 0 ≤ k ≤ 8. In addition, we show that, in asymptotic sense, the most of record Cayley graphs of diameter two are obtained by our construction.

On the restricted connectivity and superconnectivity in graphs with given girth

Discrete Mathematics, 2007

The restricted connectivity κ′(G)κ′(G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex uu has all its neighbors in X; the superconnectivity κ1(G)κ1(G) is defined similarly, this time considering only vertices uu in G-XG-X, hence κ1(G)⩽κ′(G)κ1(G)⩽κ′(G). The minimum edge-degree of G is ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, d(u)d(u) standing for the degree of a vertex uu. In this paper, several sufficient conditions yielding κ1(G)⩾ξ(G)κ1(G)⩾ξ(G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17–31] and guaranteeing κ1(G)=κ′(G)=ξ(G)κ1(G)=κ′(G)=ξ(G) under some additional constraints.

On the connectivity of -cages of even girth

Discrete Mathematics, 2008

A (k,g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper we give a brief overview of the current results on the connectivity of (k,g)-cages and we improve the current known best lower bound on the vertex connectivity of (k,g)-cages for g even.

Extraconnectivity of graphs with large minimum degree and girth

Discrete Mathematics, 1997

The extraconnectivity κ(n) of a simple connected graph G is a kind of conditional connectivity which is the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has more than n vertices. The usual connectivity and superconnectivity of G correspond to κ(0) and κ(1) respectively. This paper gives sufficient conditions, relating the diameter D, the girth g, and the minimum degree δ of a graph, to assure maximum extraconnectivity. For instance, if D ≤ g − n + 2(δ − 3), being n ≥ 2δ + 4 and g ≥ n + 5, then the value of κ(n) is (n+1)δ−2n, which is optimal. The corresponding edge version of this result, to assure maximum edge-extraconnectivity λ(n), is also discussed.

On the connectivity of (k, g)-cages of even girth

2008

A (k,g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper we give a brief overview of the current results on the connectivity of (k,g)-cages and we improve the current known best lower bound on the vertex connectivity of (k,g)-cages for g even.

On the girth of extremal graphs without shortest cycles

Discrete Mathematics, 2008

Let E X (ν; {C 3 , . . . , C n }) denote the set of graphs G of order ν that contain no cycles of length less than or equal to n which have maximum number of edges. In this paper we consider a problem posed by several authors: does G contain an n + 1 cycle? We prove that the diameter of G is at most n − 1, and present several results concerning the above question: the girth of G is g = n + 1 if (i) ν ≥ n + 5, diameter equal to n − 1 and minimum degree at least 3; (ii) ν ≥ 12, ν ∈ {15, 80, 170} and n = 6. Moreover, if ν = 15 we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if ν ≥ 2n − 3 and n ≥ 7 the girth is at most 2n − 5. We also show that the answer to the question is negative for ν ≤ n + 1 + (n − 2)/2 .