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Upper Bounds on the Order of Cages

F. Lazebnik, V. A. Ustimenko, A. J. Woldar
1996 Electronic Journal of Combinatorics  
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.  ...  The problem of determining $ {v(k,g)}$ is unsolved for most pairs $ {(k,g)}$ and is extremely hard in the general case.  ...  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.  ... 
doi:10.37236/1328 fatcat:opqfijgmifcuvixtkzydwcyiiu

Page 2898 of Mathematical Reviews Vol. , Issue 2001E [page]

2001 Mathematical Reviews  
-coloring of planar graphs with large odd-girth.  ...  Graph Theory 33 (2000), no. 2, 109-119. Summary: “The odd-girth of a graph is the length of a shortest odd circuit.  ... 

Page 39 of Mathematical Reviews Vol. , Issue 95a [page]

1995 Mathematical Reviews  
It is shown to be an NP-hard problem in general, and a linear time algorithm is presented for the class of trees.” 95a:05057 05C35 Zhang, Guo-Hui Extremal regular graphs with prescribed odd girth.  ...  Summary: “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.  ... 

On the homogeneous algebraic graphs of large girth and their applications

T. Shaska, V. Ustimenko
2009 Linear Algebra and its Applications  
We consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory.  ...  Families of finite graphs of large girth were introduced in classical extremal graph theory.  ...  We consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory.  ... 
doi:10.1016/j.laa.2008.08.023 fatcat:dlt53j7ywzcq5enffof4age6lu

Page 828 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
graphs without odd cycles of prescribed lengths.  ...  (English summary) Graphs Combin. 18 (2002), no. 1, 75-92. The odd girth g,(G) of a graph G is the length of the smallest odd cycle in the graph.  ... 

On cages with given degree sets

Denis Hanson, Ping Wang, Leif K. Jørgensen
1992 Discrete Mathematics  
Jargensen, On cages with given degree sets, Discrete Mathematics 101 (1992) 109-114. We consider the problem of constructing minimal graphs of given girth having a particular degree set.  ...  Cages have been generalized in a number of ways such as prescribing the minimum 'odd and even girth'.  ...  Introduction A (v, g)-cage (or cage) is a v-regular (simple) graph of girth g on the minimum number of vertices.  ... 
doi:10.1016/0012-365x(92)90596-8 fatcat:3le4j37s6bdf7asa5nk675k3fe

Explicit construction of graphs with an arbitrary large girth and of large size

Felix Lazebnik, Vasiliy A. Ustimenko
1995 Discrete Applied Mathematics  
Let k ~ 3 be a positive odd integer and q be a power of a prime. In this paper we give an explicit construction of a q-regular b/partite graph on r = 2q k vertices with girth 0 ~ k + 5.  ...  For any positive integer t we also give an example ofa q ~-2r-regular bipartite graph on r = 2q ~+ t vertices with girth g ~ k + 5 which is both vertex-transitive and edge-transitive.  ...  Kantor who found a mistake in the computation of girth in a preliminary construction of a D(5, q)-Iike graph and made a number of useful remarks. We are also very grateful to Professors G.A.  ... 
doi:10.1016/0166-218x(94)00058-l fatcat:nrff3b55eje3vkdpwbcexhi3au

Page 32 of Mathematical Reviews Vol. , Issue 86a [page]

1986 Mathematical Reviews  
Robert Girse (Pocatello, Idaho) Bauer, Douglas (1-STIT) 86a:05071 Extremal nonbipartite regular graphs of girth 4. J. Combin. Theory Ser. B 37 (1984), no. 1, 64-69.  ...  The reviewer’s paper was principally concerned with the number f(r, 2), defined to be the smallest number of points in a nonbipar- tite r-regular graph of girth 4 and diameter 2. J.  ... 

Moore graphs and cycles are extremal graphs for convex cycles [article]

Jernej Azarija, Sandi Klavžar
2012 arXiv   pre-print
Let ρ(G) denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 <= g <=n.  ...  The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.  ...  The second author is also with the Institute of Mathematics, Physics and Mechanics, Ljubljana.  ... 
arXiv:1210.6342v1 fatcat:bcifrjyg75fxld356m4xfh7f2m

Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles

Jernej Azarija, Sandi Klavžar
2014 Journal of Graph Theory  
Let ρ(G) denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 ≤ g ≤ n.  ...  The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.  ...  The second author is also with the Institute of Mathematics, Physics and Mechanics, Ljubljana.  ... 
doi:10.1002/jgt.21837 fatcat:pwwrjqof2vb5dhbxnkhxmg7jdy

Linear programming bounds for regular graphs [article]

Hiroshi Nozaki
2015 arXiv   pre-print
The known graphs satisfying g>2d-1 are Moore graphs, incidence graphs of regular generalized polygons of order (s,s), triangle-free strongly regular graphs, and the odd graph of degree 4.  ...  In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues.  ...  Brouwer and Haemers [12] proved that a graph with the spectrum of a distance-regular graph with diameter D and girth at least 2D − 1, is such a graph.  ... 
arXiv:1407.4562v2 fatcat:l2forhw225ab5b2kzae7f2bmzy

Linear Programming Bounds for Regular Graphs

Hiroshi Nozaki
2015 Graphs and Combinatorics  
The known graphs satisfying g > 2d − 1 are Moore graphs, incidence graphs of regular generalized polygons of order (s, s), triangle-free strongly regular graphs, and the odd graph of degree 4.  ...  In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues.  ...  Brouwer and Haemers [12] proved that a graph with the spectrum of a distance-regular graph with diameter D and girth at least 2D − 1, is such a graph.  ... 
doi:10.1007/s00373-015-1613-7 fatcat:yi2xprr6pvhrvli7ycxo5eouba

Page 1818 of Mathematical Reviews Vol. , Issue 85e [page]

1985 Mathematical Reviews  
From the introduction: “Our principal area of interest is the girth of trivalent Cayley graphs, in particular the search for the smallest Cayley graph with a given girth.”  ...  There are several results concerning path systems with prescribed ends and prescribed lengths (modulo some natural number) in graphs of sufficiently high connectivity; these strengthen results of H.  ... 

Girth, oddness, and colouring defect of snarks [article]

Ján Karabáš and Edita Máčajová and Roman Nedela and Martin Škoviera
2022 arXiv   pre-print
At the same time, our result improves Kochol's original construction of snarks with large girth (1996) in that it provides infinitely many nontrivial snarks of any prescribed girth g≥ 5, not just girth  ...  This result is achieved by means of a construction of cyclically 5-edge-connected snarks with oddness 2 and arbitrarily large girth.  ...  Oddness, girth and colouring defect In this section we discuss relationships between several measures of uncolourabilty of cubic graphs (in the sense of the survey [3] ), with particular emphasis on oddness  ... 
arXiv:2106.12205v3 fatcat:yod7bzzwafhp7gs4ykn34446xu

Graphs of Prescribed Girth and Bi-Degree

Z. Furedi, F. Lazebnik, A. Seress, V.A. Ustimenko, A.J. Woldar
1995 Journal of combinatorial theory. Series B (Print)  
Γ is called an (r, s, t)-graph if, additionally, the girth of Γ is 2t. 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.  ...  Alexander Schliep, who designed computer programs which improved our understanding of the graphs D(k, q), and to Professor Carl Pomerance for helpful discussions and for providing reference [5] .  ... 
doi:10.1006/jctb.1995.1033 fatcat:2npnrv5vv5cq7ks34tut74y4v4
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