On the connectivity of (k, g)-cages of even girth (original) (raw)
Improved lower bound for the vertex connectivity of -cages
Discrete Mathematics, 2005
A ( , g)-cage is a -regular graph with girth g and with the least possible number of vertices. We prove that all ( , g)-cages are r-connected with r √ + 1 for g 7 odd. This result supports the conjecture of Fu, Huang and Rodger that all ( ; g)-cages are -connected.
Improved lower bound for the vertex connectivity of (δ; g)-cages
2005
A ( , g)-cage is a -regular graph with girth g and with the least possible number of vertices. We prove that all ( , g)-cages are r-connected with r √ + 1 for g 7 odd. This result supports the conjecture of Fu, Huang and Rodger that all ( ; g)-cages are -connected.
On the connectivity of cages with girth five, six and eight
2007
A ( , g)-cage is a regular graph of degree and girth g with the least possible number of vertices. Recently, some authors have addressed the problem of studying the connectivity of cages. In this direction, it was conjectured by Fu, Huang and Rodger that every ( , g)-cage is maximally connected, i.e., it is -connected, and they proved this statement for = 3. We provide a new contribution to the proof of that conjecture, by showing that every ( , g)-cage with g = 6, 8 is maximally connected, and by assuring either maximal connectivity or superconnectivity for some ( , 5)-cages.
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.
All (k;g)-cages arek-edge-connected
Journal of Graph Theory, 2005
A (k;g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper, we prove that (k;g)-cages are k-edge-connected if g is even. Earlier, Wang, Xu, and Wang proved that (k;g)-cages are k-edge-connected if g is odd. Combining our results, we conclude that the (k;g)-cages are k-edge-connected. ß
A new bound for the connectivity of cages
2012
An (r, g)-cage is an r-regular graph of girth g of minimum order. We prove that all (r, g)cages are at least r/2 -connected for every odd girth g ≥ 7 by means of a matrix technique which allows us to construct graphs without short cycles. This lower bound on the vertex connectivity of cages is a new advance in proving the conjecture of Fu, Huang and Rodger which states that all (r, g)-cages are r-connected.
Journal of Graph Theory, 1997
A (k; g)-graph is a k-regular graph with girth g. Let f (k; g) be the smallest integer ν such there exists a (k; g)-graph with ν 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; g 1) < f(k; g 2) for all k ≥ 3 and 3 ≤ g 1 < g 2. We use this to prove that (k; g)-cages are 2-connected, and if k = 3 then their connectivity is k.
Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07, 2007
Constructing some regular graph with a given girth, a given degree and the fewest possible vertices is a hard problem. This problem is called the cage graph problem and has some links with the error codes theory. In this paper we presents some new graphs, constructed from a group, with a girth of 6 and regular of degree p, for any prime number p. This graphs are of order 2×p 2 when the best upper bound known for the (p, 6)-cage problem was the Sauer bound, equal to 4(p − 1) 3 .
G-graphs for the cage problem: a new upper bound
International Symposium on Symbolic and Algebraic Computation, 2007
Constructing some regular graph with a given girth, a given degree and the fewest possible vertices is a hard problem. This problem is called the cage graph problem and has some links with the error codes theory. In this paper we presents some new graphs, constructed from a group, with a girth of 6 and regular of degree p, for any prime number p. This graphs are of order 2×p 2 when the best upper bound known for the (p, 6)-cage problem was the Sauer bound, equal to 4(p − 1) 3 .
New family of small regular graphs of girth 5
Electronic Notes in Discrete Mathematics, 2016
A (k, g)-cage is a k-regular graph of girth g of minimum order. In this work, we focus on girth g = 5, where cages are known only for degrees k ≤ 7. When k ≥ 8, except perhaps for k = 57, the order of a (k, 5)-cage is strictly greater
(Delta, G)-Cages with G >= 10 Are 4-CONNECTED
Discrete Mathematics, 2005
A regular graph G of degree and girth g is said to be a (, g)-cage if it has the least number of vertices among all-regular graphs with girth g. A graph is called k-connected if the order of every cutset is at least k. In this work, we prove that every (, g)-cage is 4-connected provided that either = 4, or 5 and g 10. These results support the conjecture of Fu, Huang and Rodger that all (, g)-cages are-connected.
New Upper Bounds on the Order of CAGES1
The Electronic Journal of Combinatorics, 1996
Let k≥2 and g≥3 be integers. A (k,g)-graph is a k-regular graph with girth (length of a smallest cycle) exactly g. A (k,g)-cage is a (k,g)-graph of minimum order. Let v(k,g) be the order of a (k,g)-cage. The problem of determining v(k,g) is unsolved for most pairs (k,g) and is extremely hard in the general case. It is easy to establish the following lower bounds for v(k,g): v(k,g)≥ k(k−1) (g−1)/2 −2 k−2 for g odd, and v(k,g)≥ 2(k−1) g/2 −2 k−2 for g even. The best known upper bounds are roughly the squares of the lower bounds. In this paper we establish general upper bounds on v(k,g) which are roughly the 3/2 power of the lower bounds, and we provide explicit constructions for such (k,g)-graphs.
Discrete Mathematics, 2005
A regular graph G of degree and girth g is said to be a ( , g)-cage if it has the least number of vertices among all -regular graphs with girth g. A graph is called k-connected if the order of every cutset is at least k. In this work, we prove that every ( , g)-cage is 4-connected provided that either = 4, or 5 and g 10. These results support the conjecture of Fu, Huang and Rodger that all ( , g)-cages are -connected.
(δ, g)-cages with g⩾ 10 are 4-connected
2005
A regular graph G of degree and girth g is said to be a ( , g)-cage if it has the least number of vertices among all -regular graphs with girth g. A graph is called k-connected if the order of every cutset is at least k. In this work, we prove that every ( , g)-cage is 4-connected provided that either = 4, or 5 and g 10. These results support the conjecture of Fu, Huang and Rodger that all ( , g)-cages are -connected.
Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages
Let q be a prime power and g ∈ {6, 8, 12}. In this paper we obtain (q, g)graphs on 2q g/2−3 (q 2 −1) vertices for g = 6, 8, 12 as subgraphs of known (q +1, g)cages. We also obtain (k, 6)-graphs on 2(kq − 1) vertices, and (k, 8)-graphs on 2k(q 2 −1) vertices and (k, 12)-graphs on 2kq 2 (q 2 −1), where k is a positive integer * garaujo@math.unam.mx † m.camino.balbuena@upc.edu ‡ hetamas@cs.elte.hu such that q ≥ k ≥ 3. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 6, 8, 12.
Improving bounds on the order of regular graphs of girth 5
Discrete Mathematics, 2019
A (k, g)-graph is a k-regular graph with girth g and a (k, g)-cage is a (k, g)-graph with the fewest possible number of vertices n(k, g). Constructing (k, g)-cages and determining the order are both very hard problems. For this reason, an intensive line of research is devoted to constructing smaller (k, g)-graphs than previously known ones, providing in this way new upper bounds to n(k, g) each time such a graph is constructed. The paper focuses on girth g = 5, where cages are known only for degrees k ≤ 7. We construct (k, 5)-graphs using and extending techniques of amalgamation into the incidence graphs of elliptic semiplanes of type L introduced and exposed by . The order of these graphs provides better upper bounds on n(k, 5) than those known so far, for values of k such that either 13 ≤ k ≤ 33 or k ≥ 66.
New families of small regular graphs of girth 5
2015
In this paper we are interested in the Cage Problem that consists in constructing regular graphs of given girth g and minimum order. We focus on girth g=5, where cages are known only for degrees k < 7. We construct regular graphs of girth 5 using techniques exposed by Funk [Note di Matematica. 29 suppl.1, (2009) 91 - 114] and Abreu et al. [Discrete Math. 312 (2012), 2832 - 2842] to obtain the best upper bounds known hitherto. The tables given in the introduction show the improvements obtained with our results.
Edge-connectivity in Pk-path graphs
2004
The P k (G)-path graph corresponding to a graph G has for vertices the set of all paths of length k in G. Two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k − 1 in G, and their union forms either a cycle or a path of length k + 1. Path graphs were introduced by Broersma and Hoede (J. Graph. Theory 13 as a generalization of line graphs, because for k = 1, path graphs are just line graphs. Results on the edge-connectivity of line graphs are given by Chartrand and Stewart (12). The connectivity of P k -path graphs has been studied by Knor and Niepel (Graph Theory 20 (2000) 181), where they proved a necessary and sufficient condition for the P k (G)-path graphs to be disconnected, assuming that G has girth of at least k + 1. Going one step further, we prove in this work that the edge-connectivity of P k (G) is at least (P k (G)) (G) − 1 for a graph G of girth at least k + 1 and minimum degree (G) 2. Furthermore, we show (P k (G)) 2 (G) − 2 provided that (G) 3.