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Upper Bounds on the Order of Cages

1996
*
Electronic Journal of Combinatorics
*

A $ {(k,g)}$-

doi:10.37236/1328
fatcat:opqfijgmifcuvixtkzydwcyiiu
*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. ...##
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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. ...##
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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. ...##
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On the homogeneous algebraic graphs of large girth and their applications

2009
*
Linear Algebra and its Applications
*

We consider the problem of constructing homogeneous algebraic

doi:10.1016/j.laa.2008.08.023
fatcat:dlt53j7ywzcq5enffof4age6lu
*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. ...##
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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*. ...

##
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On cages with given degree sets

1992
*
Discrete Mathematics
*

Jargensen, On cages

doi:10.1016/0012-365x(92)90596-8
fatcat:3le4j37s6bdf7asa5nk675k3fe
*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. ...##
###
Explicit construction of graphs with an arbitrary large girth and of large size

1995
*
Discrete Applied Mathematics
*

Let k ~ 3 be a positive

doi:10.1016/0166-218x(94)00058-l
fatcat:nrff3b55eje3vkdpwbcexhi3au
*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. ...##
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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. ...##
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Moore graphs and cycles are extremal graphs for convex cycles
[article]

2012
*
arXiv
*
pre-print

Let ρ(G) denote the number of convex cycles of a simple

arXiv:1210.6342v1
fatcat:bcifrjyg75fxld356m4xfh7f2m
*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. ...##
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Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles

2014
*
Journal of Graph Theory
*

Let ρ(G) denote the number of convex cycles of a simple

doi:10.1002/jgt.21837
fatcat:pwwrjqof2vb5dhbxnkhxmg7jdy
*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. ...##
###
Linear programming bounds for regular graphs
[article]

2015
*
arXiv
*
pre-print

The known

arXiv:1407.4562v2
fatcat:l2forhw225ab5b2kzae7f2bmzy
*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*. ...##
###
Linear Programming Bounds for Regular Graphs

2015
*
Graphs and Combinatorics
*

The known

doi:10.1007/s00373-015-1613-7
fatcat:yi2xprr6pvhrvli7ycxo5eouba
*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*. ...##
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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]

2022
*
arXiv
*
pre-print

At the same time, our result improves Kochol's original construction of snarks

arXiv:2106.12205v3
fatcat:yod7bzzwafhp7gs4ykn34446xu
*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*...##
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Graphs of Prescribed Girth and Bi-Degree

1995
*
Journal of combinatorial theory. Series B (Print)
*

Γ is called an (r, s, t)-

doi:10.1006/jctb.1995.1033
fatcat:2npnrv5vv5cq7ks34tut74y4v4
*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] . ...
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