Bounds on the order of biregular graphs with even girth at least 8 (original) (raw)

On bi-regular cages of even girth at least 8

Aequationes mathematicae, 2013

Let r, m, 2 ≤ r < m and g ≥ 3 be three positive integers. A graph with a prescribed degree set r, m and girth g having the least possible number of vertices is called a biregular cage or an (r, m; g)-cage, and its order is denoted by n(r, m; g). In this paper we provide upper bounds on n(r, m; g) for some related values of r, m and even girth g at least 8. Moreover, if r − 1 is a prime power and m ≥ 5, we construct the smallest currently known (r, m; 8)-graphs. Also, if r = 3 and m ≥ 7 is not divisible by 3, we prove that n(3, m; 8) ≥ ⌈25m/3⌉ + 7. Finally, we construct a family of (3, m; 8)-graphs of order 9m + 3 which are cages for m = 4, 5, 7.

On the order of bi-regular cages of even girth

2006

By a bi-regular cage of girth g we mean a graph with prescribed degrees r and m and with the least possible number of vertices denoted by f ({r, m}; g). We provide new upper and lower bounds of f ({r, m}; g) for even girth g ≥ 6. Moreover, we prove that f ({r, k(r − 1) + 1}; 6) = 2k(r − 1) 2 + 2r where k ≥ 2 is any integer and r − 1 is a prime power. This result supports the conjecture f ({r, m}; 6) = 2(rm − m + 1) for any r < m formulated by Yuansheng and Liang (The minimum number of vertices with girth 6 and degree set D = {r, m}, Discrete Mathematics 269 , 249-258).

On the order of ({r, m};g)-cages of even girth

Discrete Mathematics, 2008

By an ({r,m};g)({r,m};g)-cage we mean a graph on a minimum number of vertices f({r,m};g)f({r,m};g) with degree set {r,m}{r,m}, 2⩽r<m2⩽r<m, and girth g. In this paper we improve the known lower bound for f({r,m};g)f({r,m};g) for even girth g⩾8g⩾8. Moreover, we obtain for any integer k⩾2k⩾2 that f({r,k(r-1)+1};6)=2k(r-1)2+2rf({r,k(r-1)+1};6)=2k(r-1)2+2r where r-1r-1 a is prime power. This result supports the conjecture that f({r,m};6)=2(rm-m+1)f({r,m};6)=2(rm-m+1) for any r<mr<m posed by Yuansheng and Liang [The minimum number of vertices with girth 6 and degree set D={r,m}D={r,m}, Discrete Math. 269 (2003) 249–258].

New upper bounds on the order of cages

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 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.

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.

Upper Bounds on the Order of Cages

The Electronic Journal of Combinatorics

Let $ { k\ge 2}$ and $ {g\ge3}$ 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)\ge} {{k(k-1)^{(g-1)/2}-2}\over {k-2}}$ for $ {g}$ odd, and $ {v(k,g)\ge} { {2(k-1)^{g/2}-2}\over {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.

On upper bounds of odd girth cages

Discrete Mathematics, 2010

We give a construction of k-regular graphs of girth g using only geometrical and combinatorial properties that appear in any (k; g + 1)-cage, a minimal k-regular graph of girth g + 1. In this construction, g ≥ 5 and k ≥ 3 are odd integers, in particular when k − 1 is a power of 2 and (g + 1) ∈ {6, 8, 12} we use the structure of generalized polygons. With this construction we obtain upper bounds for the (k; g)-cages. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 5, 7, 11.

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 .

Constructions of bi-regular cages

Discrete Mathematics, 2009

Given three positive integers r, m and g, one interesting question is the following: What is the minimum number of vertices that a graph with prescribed degree set {r, m}, 2 ≤ r < m, and girth g can have? Such a graph is called a bi-regular cage or an ({r, m}; g)-cage, and its minimum order is denoted by n({r, m}; g). In this paper we provide new upper bounds on n({r, m}; g) for some related values of r and m. Moreover, if r − 1 is a prime power, we construct the following bi-regular cages: ({r, k(r − 1)}; g)-cages for g ∈ {5, 7, 11} and k ≥ 2 even; and ({r, kr}; 6)-cages for k ≥ 2 any integer. The latter cages are of order n({r, kr}; 6) = 2(kr 2 − kr + 1). Then this result supports the conjecture that n({r, m}; 6) = 2(rm − m + 1) for any r < m, posed by Yuansheng and Liang [The minimum number of vertices with girth 6 and degree set D = {r, m}, Discrete Mathematics 269 , 249-258]. We finalize giving the exact value n({3, 3k}; 8), for k ≥ 2.

A construction of small regular bipartite graphs of girth 8

2009

Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq 2 − q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.