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Large-girth roots of graphs
[article]

2009
*
arXiv
*
pre-print

We study the problem

arXiv:0909.4011v1
fatcat:4bhqwi4czjgidktsprctqvvd5y
*of*recognizing*graph*powers and computing*roots**of**graphs*. ... Our algorithm also finds all r-th*roots**of*a given*graph*that have*girth*at least 2r+3 and no degree one vertices, which is a step towards a recent conjecture*of*Levenshtein that such*root*should be unique ... In this work we address the above problems for another*large*family*of**graphs*, namely*graphs*with no short cycles. Recall, that the*girth**of*a*graph*is the length*of*its shortest cycle. ...##
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Large-Girth Roots of Graphs

2010
*
SIAM Journal on Discrete Mathematics
*

We study the problem

doi:10.1137/100792949
fatcat:aqgpi37h5faihn6wvxodl624ay
*of*recognizing*graph*powers and computing*roots**of**graphs*. ... Our algorithm also finds all r-th*roots**of*a given*graph*that have*girth*at least 2r + 3 and no degree one vertices, which is a step towards a recent conjecture*of*Levenshtein that such*root*should be ... In this work we address the above problems for another*large*family*of**graphs*, namely*graphs*with no short cycles. ...##
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Some voltage graph-based LDPC tailbiting codes with large girth

2011
*
2011 IEEE International Symposium on Information Theory Proceedings
*

The relation between the parity-check matrices

doi:10.1109/isit.2011.6034230
dblp:conf/isit/BocharovaHJKS11
fatcat:pjvvghjaefbcllyoprrieabrpm
*of*quasi-cyclic (QC) low-density parity-check (LDPC) codes and the biadjacency matrices*of*bipartite*graphs*supports searching for powerful LDPC block codes ... Algorithms for searching iteratively for LDPC block codes with*large**girth*are presented and constructions based on Steiner Triple Systems and short QC block codes are introduced, leading to new QC regular ... The problem*of*finding QC LDPC codes with*large**girth*was considered in several papers. ...##
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On the homogeneous algebraic graphs of large girth and their applications

2009
*
Linear Algebra and its Applications
*

Families

doi:10.1016/j.laa.2008.08.023
fatcat:dlt53j7ywzcq5enffof4age6lu
*of*finite*graphs**of**large**girth*were introduced in classical extremal*graph*theory. ... We consider some results on such algebraic*graphs*over any field. The upper bound on the dimension*of*variety*of*edges for algebraic*graphs**of**girth*2d is established. ... and families*of**graphs**of**large**girth*(see Section 2). ...##
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Graphs with bounded tree-width and large odd-girth are almost bipartite
[article]

2009
*
arXiv
*
pre-print

We prove that for every k and every ε>0, there exists g such that every

arXiv:0904.2282v1
fatcat:lhiqcecp5vdvzbvz646yyoyeae
*graph*with tree-width at most k and odd-*girth*at least g has circular chromatic number at most 2+ε. ... More precisely, as Thomassen observed [8] , a*graph*that avoids a fixed minor and has*large**girth*is 2-degenerate, and hence 3-colorable. ... On the other hand,*graphs*with*large**girth*that avoid a fixed minor are known to have low chromatic number (in particular, this applies to*graphs*embedded on a fixed surface). ...##
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Dense Minors In Graphs Of Large Girth

2004
*
Combinatorica
*

As a corollary, every

doi:10.1007/s00493-005-0009-3
fatcat:5twzumbpf5eylmkdt7gkuxebxi
*graph**of**girth*at least 6 log r + 3 log log r + c and minimum degree at least 3 has a Kr minor. ... We show that a*graph**of**girth*greater than 6 log k + 3 and minimum degree at least 3 has a minor*of*minimum degree greater than k. This is best possible up to a factor*of*at most 9/4. ... The best lower bound we have found is 8 3 log k − c, but we note that existing conjectures about cubic*graphs**of**large**girth*would raise this to about 4 log k. ...##
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Multicolour Ramsey Numbers of Odd Cycles
[article]

2017
*
arXiv
*
pre-print

We use these colourings to give new lower bounds on the k-colour Ramsey number

arXiv:1602.07607v2
fatcat:wo6oectounhxtctedfqv2tzzqu
*of*the odd cycle and prove that, for all odd r and all k sufficiently*large*, there exists a constant ϵ = ϵ(r) > 0 such that ... This makes progress on a problem*of*Erdős and Graham and answers a question*of*Chung. ... We say that an r-round*graph*G is*rooted*with*root*O, for some vertex O ∈ V (G), if X 1 = {O}. ...##
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Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth

2012
*
IEEE Transactions on Information Theory
*

Algorithms for searching iteratively for LDPC block codes with

doi:10.1109/tit.2011.2176717
fatcat:xfazir3i4rd5bezdek5l5eqyau
*large**girth*and for determining their minimum distance are presented. ... Using the principle*of*tailbiting, compact representations*of*bipartite*graphs*based on convolutional codes can be found. ... SEARCHING FOR QC LDPC BLOCK CODES WITH*LARGE**GIRTH*When searching for QC LDPC block codes with*large**girth*, we start from a base*graph**of*a rate R = b/c (J, K)-regular LDPC convolutional code. ...##
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Graphs with bounded tree-width and large odd-girth are almost bipartite

2010
*
Journal of combinatorial theory. Series B (Print)
*

We prove that for every k and every ε > 0, there exists g such that every

doi:10.1016/j.jctb.2010.04.004
fatcat:ujqe4fpa75bjjm5bbbi2u4azcy
*graph*with tree-width at most k and odd-*girth*at least g has circular chromatic number at most 2 + ε. ... More precisely, as Thomassen observed [8] , a*graph*that avoids a fixed minor and has*large**girth*is 2degenerate, and hence 3-colorable. ... On the other hand,*graphs*with*large**girth*that avoid a fixed minor are known to have low chromatic number (in particular, this applies to*graphs*embedded on a fixed surface). ...##
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Multicolour Ramsey numbers of odd cycles

2017
*
Journal of combinatorial theory. Series B (Print)
*

We use these colourings to give new lower bounds on the k-colour Ramsey number

doi:10.1016/j.jctb.2016.12.005
fatcat:gfcqi3pzovhq3bjzze3wpauczq
*of*the odd cycle and prove that, for all odd r and all k sufficiently*large*, there exists a constant = (r) > 0 such that R ... This makes progress on a problem*of*Erdős and Graham and answers a question*of*Chung. ... We say that an r-round*graph*G is*rooted*with*root*O, for some vertex O ∈ V (G), if X 1 = {O}. ...##
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Tree-like distance colouring for planar graphs of sufficient girth
[article]

2019
*
arXiv
*
pre-print

sufficiently

arXiv:1805.02156v2
fatcat:ea4s3t6jnjahvh5nv6vu62g6gu
*large*. ... We prove for odd t the existence*of*a quantity g depending only on t such that the distance-t chromatic index*of*any planar multigraph*of*maximum degree d and*girth*at least g is at most τ'_t(d) if d is ... Acknowledgements The first author would like to thank Magnús Halldórsson for helpful discussions at a very early stage*of*this work. ...##
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Computational determination of (3,11) and (4,7) cages

2011
*
Journal of Discrete Algorithms
*

A (k, g)-

doi:10.1016/j.jda.2010.11.001
fatcat:kf7p5vltabfihezcndb6roulpe
*graph*is a k-regular*graph**of**girth*g, and a (k, g)-cage is a (k, g)-*graph**of*minimum order. We show that a (3, 11)-*graph**of*order 112 found by Balaban in 1973 is minimal and unique. ... We also show that the order*of*a (4, 7)-cage is 67 and find one example. Finally, we improve the lower bounds on the orders*of*(3, 13)-cages and (3, 14)-cages to 202 and 260, respectively. ... A (k, g)-*graph*is a regular*graph**of*degree k and*girth*g, and a (k, g)-cage is a (k, g)-*graph**of*minimum possible order. ...##
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Cycle lengths in sparse graphs
[article]

2007
*
arXiv
*
pre-print

Let C(G) denote the set

arXiv:0707.2117v1
fatcat:gved64uipnfxlnj4bdkflfdptq
*of*lengths*of*cycles in a*graph*G. In the first part*of*this paper, we study the minimum possible value*of*|C(G)| over all*graphs*G*of*average degree d and*girth*g. ... We also show that Ω(d^ (g-1)/2) is a lower bound for the number*of*odd cycle lengths in a*graph**of*chromatic number d and*girth*g. ... The authors would like to thank the referee for careful reading*of*this manuscript. ...##
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Cycle lengths in sparse graphs

2008
*
Combinatorica
*

Let C(G) denote the set

doi:10.1007/s00493-008-2300-6
fatcat:336a6ssplfff7pnu66yeiaabay
*of*lengths*of*cycles in a*graph*G. In the first part*of*this paper, we study the minimum possible value*of*|C(G)| over all*graphs*G*of*average degree d and*girth*g. ... We also show that Ω d (g−1)/2 is a lower bound for the number*of*odd cycle lengths in a*graph**of*chromatic number d and*girth*g. ... The authors would like to thank the referee for careful reading*of*this manuscript. ...##
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Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs

2014
*
Combinatorica
*

Our methods also lead to explicit estimates on the number

doi:10.1007/s00493-014-3066-7
fatcat:go6wcemlwzgbbkqwf3xpu76yrm
*of*proper colorings*of**graphs*with*large**girth*. arXiv:1201.3861v3 [math.CO] ... We define the chromatic measure*of*a finite simple*graph*as the uniform distribution on its chromatic*roots*. ... than just*large**girth*. ...
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