Connectivity of cages

A (k; g)-graph is a k-regular graph with girth g. Let f(k; g) be the smallest integer v such there exists a (k; g)-graph with v vertices. A (k; g)-cage is a (k; g)-graph with f(k; g) vertices. In this paper we prove that the cages are monotonic in that f(k; g₁) < f(k; g₂) for all k ≥ 3 and 3 ≥ g₁ < g₂. We use this to prove that (k; g)-cages are 2-connected, and if k = 3 then their connectivity is k. © 1997 John Wiley & Sons, Inc.     

Connectivity and separating sets of cages

A (k; g)-cage is a graph of minimum order among k-regular graphs with girth g. We show that for every cutset S of a (k; g)-cage G, the induced subgraph G[S] has diameter at least ⌊g/2⌋, with equality only when distance ⌊g/2⌋ occurs for at least two pairs of vertices in G[S]. This structural property is used to prove that every (k; g)-cage with k ≥ 3 is 3-connected. This result supports the conjecture of Fu, Huang, and Rodger that every (k; g)-cage is k-connected. A nonseparating g-cycle C in a graph G is a cycle of length g such that G − V(C) is connected. We prove that every (k; g)-cage contains a nonseparating g-cycle. For even g, we prove that every g-cycle in a (k; g)-cage is nonseparating. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 35–44, 1998     

Smallest regular graphs with prescribed odd girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let f(k,g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. The exact values of f(k,g) are determined if one of the following holds:

• (i) k > 2g ?5 and k is a prime number,
• (ii) k > (2?(g + 1)/4? ?1)², and
• (iiii) k is a perfect square.

     

Connectivity of graphs with given girth pair

Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209-218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g,h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g,h) such that g is odd and h?g+3 is even has high (vertex-)connectivity if its diameter is at most h-3. The edge version of all results is also studied.     

The face pair of planar graphs

Harary and Kovacs [Smallest graphs with given girth pair, Caribbean J. Math. 1 (1982) 24-26] have introduced a generalization of the standard cage question—r-regular graphs with given odd and even girth pair. The pair (ω,ε) is the girth pair of graph G if the shortest odd and even cycles of G have lengths ω and ε, respectively, and denote the number of vertices in the (r,ω,ε)-cage by f(r,ω,ε). Campbell [On the face pair of cubic planar graph, Utilitas Math. 48 (1995) 145-153] looks only at planar graphs and considers odd and even faces rather than odd and even cycles. He has shown that f(3,ω,4)=2ω and the bounds for the left cases. In this paper, we show the values of f(r,ω,ε) for the left cases where (r,ω,ε)∈{(3,3,ε),(4,3,ε),(5,3,ε), (3,5,ε)}.     

The maximum valency of regular graphs with given order and odd girth

The odd girth of a graph G is the length of a shortest odd cycle in G. Let d(n, g) denote the largest k such that there exists a k-regular graph of order n and odd girth g. It is shown that dn, g ≥ 2|n/g≥ if n ≥ 2g. As a consequence, we prove a conjecture of Pullman and Wormald, which says that there exists a 2j-regular graph of order n and odd girth g if and only if n ≥ gj, where g ≥ 5 is odd and j ≥ 2. A different variation of the problem is also discussed.     

Properties of Groups for the Cage and Degree/Diameter Problems

A (k, g)-cage is a k-regular graph of girth g of minimum order. While many of the best known constructions of small k-regular graphs of girth g are known to be Cayley graphs, no general theory of the relation between the girth of a Cayley graph and the structure of the underlying group has been developed. We attempt to fill the gap by focusing on the girths of Caley graphs of nilpotent and solvable groups, and present a series of results supporting the intuitive idea that the closer a group is to being abelian, the less suitable it is for constructing Cayley graphs of large girths. Specifically, we establish the existence of upper bounds on the girths of Cayley graphs with respect to the nilpotency class and/or the length of the derived sequence of the underlying groups.     

On Superconnectivity of (4, g)-Cages

A (k, g)-cage is a graph that has the least number of vertices among all k-regular graphs with girth g. It has been conjectured (Fu et?al. in J. Graph Theory, 24:187?C191, 1997) that all (k, g)-cages are k-connected for every k??? 3. A k-connected graph G is called superconnected if every k-cutset S is the neighborhood of some vertex. Moreover, if G?S has precisely two components, then G is called tightly superconnected. In this paper, we prove that every (4, g)-cage is tightly superconnected when g ???11 is odd.     

Distance-biregular graphs with 2-valent vertices and distance-regular line graphs

A graph G is called distance-regularized if each vertex of G admits an intersection array. It is known that every distance-regularized graph is either distance-regular (DR) or distance-biregular (DBR). Note that DBR means that the graph is bipartite and the vertices in the same color class have the same intersection array. A (k, g)-graph is a k-regular graph with girth g and with the minimum possible number of vertices consistent with these properties. Biggs proved that, if the line graph L(G) is distance-transitive, then G is either K_1,n or a (k, g)-graph. This result is generalized to DR graphs by showing that the following are equivalent: (1) L(G) is DR and G ≠ K_1,n for n ≥ 2, (2) G and L(G) are both DR, (3) subdivision graph S(G) is DBR, and (4) G is a (k, g)-graph. This result is used to show that a graph S is a DBR graph with 2-valent vertices iff S = K_2,′ or S is the subdivision graph of a (k, g)-graph. Let G⁽²⁾ be the graph with vertex set that of G and two vertices adjacent if at distance two in G. It is shown that for a DBR graph G, G⁽²⁾ is two DR graphs. It is proved that a DR graph H without triangles can be obtained as a component of G⁽²⁾ if and only if it is a (k, g)-graph with g ≥ 4.     

New improvements on connectivity of cages

A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r − 1)² ≤ δ + $ \sqrt \delta $ \sqrt \delta − 2 < r ² and all (δ, g)-cages with even girth g ≥ 10 are r-connected, where r is the largest integer satisfying $ \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta $ \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta . These results support a conjecture of Fu, Huang and Rodger that all (δ, g)-cages are δ-connected.     

On k-ordered Hamiltonian graphs

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = − 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k − 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) > g(k, n) if 2k ≤ n ≤ 3k − 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999     

A family of regular graphs of girth 5

Murty [A generalization of the Hoffman-Singleton graph, Ars Combin. 7 (1979) 191-193.] constructed a family of (p^m+2)-regular graphs of girth five and order 2p^2m, where p?5 is a prime, which includes the Hoffman-Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497-504]. This construction gives an upper bound for the least number f(k) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f(k), k?16.     

The non-existence of certain regular graphs of girth 5

For certain positive integers k it is shown that there is no k-regular graph with girth 5 having k² + 3 vertices. This provides a new lower bound for the number of vertices of girth 5 graphs with these valences.     

Extremal Regular Graphs with Prescribed Odd Girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let ƒ(k, g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. It is known that ƒ(k, g) ≥ kg/2 and that ƒ(k, g) = kg/2 if k is even. The exact values of ƒ(k, g) are also known if k = 3 or g = 5. Let x_e denote the smallest even integer no less than x, δ(g) = (−1)^{g − 1}/2, and s(k) = min {p + q | k = pq, where p and q are both positive integers}. It is proved that if k ≥ 5 and g ≥ 7 are both odd, then [formula] with the exception that ƒ(5, 7) = 20.     

On Narrow Hexagonal Graphs with a 3-Homogeneous Suborbit

A connected graph of girth m 3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showed that the study of such graphs divides naturally into a number of substantial subcases. Here we analyze one of these cases and characterize the k-valent polygonal graphs of girth 6 which have automorphism groups transitive on vertices, which preserve the set of special hexagons, and which have a suborbit of size k – 1 at distance three from a given vertex.     

Odd‐angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ?: G□ H→S□T is a graph homomorphism, then either ? maps G□{h} to S□{t_h} for all h∈V(H) where t_h∈V(T) depends on h; or ? maps G□{h} to {s_h}□ T for all h∈V(H) where s_h∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008     

Cayley Cages

A (k,g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order. We present an explicit construction of (k,g)-Cayley graphs for all parameters k≥2 and g≥3 and generalize this construction to show that many well-known small k-regular graphs of girth g can be constructed in this way. We also establish connections between this construction and topological graph theory, and address the question of the order of (k,g)-Cayley cages.     

Phase transition of the minimum degree random multigraph process

We study the phase transition of the minimum degree multigraph process. We prove that for a constant h_g ≈︁ 0.8607, with probability tending to 1 as n → ∞, the graph consists of small components on O(log n) vertices when the number of edges of a graph generated so far is smaller than h_gn, the largest component has order roughly n^2/3 when the number of edges added is exactly h_gn, and the graph consists of one giant component on Θ(n) vertices and small components on O(log n) vertices when the number of edges added is larger than h_gn. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Subgroup-Chain Graphs

Starting from chains of subgroups in a group a construction for k-regular graphs is given and as an application of this construction a cubic graph of girth 20 with 6072 vertices is produced. This graph is exceptional in that, at the time of writing, it has fewest vertices for a cubic graph of this girth.     

The number of spanning trees in regular graphs

Let C(G) denote the number of spanning trees of a graph G. It is shown that there is a function ?(k) that tends to zero as k tends to infinity such that for every connected, k-regular simple graph G on n vertices C(G) = {k[1 ? δ(G)]}ⁿ. where 0 ≤ δ(G) ≤ ?(k).