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Large-girth roots of graphs [article]

Anna Adamaszek, Michal Adamaszek
2009 arXiv   pre-print
We study the problem 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.  ... 
arXiv:0909.4011v1 fatcat:4bhqwi4czjgidktsprctqvvd5y

Large-Girth Roots of Graphs

Anna Adamaszek, MichaŁ Adamaszek
2010 SIAM Journal on Discrete Mathematics  
We study the problem 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.  ... 
doi:10.1137/100792949 fatcat:aqgpi37h5faihn6wvxodl624ay

Some voltage graph-based LDPC tailbiting codes with large girth

Irina E. Bocharova, Florian Hug, Rolf Johannesson, Boris D. Kudryashov, Roman V. Satyukov
2011 2011 IEEE International Symposium on Information Theory Proceedings  
The relation between the parity-check matrices 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.  ... 
doi:10.1109/isit.2011.6034230 dblp:conf/isit/BocharovaHJKS11 fatcat:pjvvghjaefbcllyoprrieabrpm

On the homogeneous algebraic graphs of large girth and their applications

T. Shaska, V. Ustimenko
2009 Linear Algebra and its Applications  
Families 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).  ... 
doi:10.1016/j.laa.2008.08.023 fatcat:dlt53j7ywzcq5enffof4age6lu

Graphs with bounded tree-width and large odd-girth are almost bipartite [article]

Alexandr V. Kostochka, Daniel Kral', Jean-Sebastien Sereni, Michael Stiebitz
2009 arXiv   pre-print
We prove that for every k and every ε>0, there exists g such that every 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).  ... 
arXiv:0904.2282v1 fatcat:lhiqcecp5vdvzbvz646yyoyeae

Dense Minors In Graphs Of Large Girth

Reinhard Diestel, Christof Rempel
2004 Combinatorica  
As a corollary, every 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.  ... 
doi:10.1007/s00493-005-0009-3 fatcat:5twzumbpf5eylmkdt7gkuxebxi

Multicolour Ramsey Numbers of Odd Cycles [article]

A. Nicholas Day, J. Robert Johnson
2017 arXiv   pre-print
We use these colourings to give new lower bounds on the k-colour Ramsey number 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}.  ... 
arXiv:1602.07607v2 fatcat:wo6oectounhxtctedfqv2tzzqu

Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth

Irina E. Bocharova, Florian Hug, Rolf Johannesson, Boris D. Kudryashov, Roman V. Satyukov
2012 IEEE Transactions on Information Theory  
Algorithms for searching iteratively for LDPC block codes with 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.  ... 
doi:10.1109/tit.2011.2176717 fatcat:xfazir3i4rd5bezdek5l5eqyau

Graphs with bounded tree-width and large odd-girth are almost bipartite

Alexandr V. Kostochka, Daniel Král', Jean-Sébastien Sereni, Michael Stiebitz
2010 Journal of combinatorial theory. Series B (Print)  
We prove that for every k and every ε > 0, there exists g such that every 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).  ... 
doi:10.1016/j.jctb.2010.04.004 fatcat:ujqe4fpa75bjjm5bbbi2u4azcy

Multicolour Ramsey numbers of odd cycles

A. Nicholas Day, J. Robert Johnson
2017 Journal of combinatorial theory. Series B (Print)  
We use these colourings to give new lower bounds on the k-colour Ramsey number 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}.  ... 
doi:10.1016/j.jctb.2016.12.005 fatcat:gfcqi3pzovhq3bjzze3wpauczq

Tree-like distance colouring for planar graphs of sufficient girth [article]

Ross J. Kang, Willem van Loon
2019 arXiv   pre-print
sufficiently 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.  ... 
arXiv:1805.02156v2 fatcat:ea4s3t6jnjahvh5nv6vu62g6gu

Computational determination of (3,11) and (4,7) cages

Geoffrey Exoo, Brendan D. McKay, Wendy Myrvold, Jacqueline Nadon
2011 Journal of Discrete Algorithms  
A (k, g)-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.  ... 
doi:10.1016/j.jda.2010.11.001 fatcat:kf7p5vltabfihezcndb6roulpe

Cycle lengths in sparse graphs [article]

Benny Sudakov, Jacques Verstraete
2007 arXiv   pre-print
Let C(G) denote the set 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.  ... 
arXiv:0707.2117v1 fatcat:gved64uipnfxlnj4bdkflfdptq

Cycle lengths in sparse graphs

Benny Sudakov, Jacques Verstraëte
2008 Combinatorica  
Let C(G) denote the set 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.  ... 
doi:10.1007/s00493-008-2300-6 fatcat:336a6ssplfff7pnu66yeiaabay

Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs

Miklós Abért, Tamás Hubai
2014 Combinatorica  
Our methods also lead to explicit estimates on the number 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.  ... 
doi:10.1007/s00493-014-3066-7 fatcat:go6wcemlwzgbbkqwf3xpu76yrm
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