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Graphs with large maximum degree containing no odd cycles of a given length

Paul Balister, Béla Bollobás, Oliver Riordan, Richard H. Schelp
<span title="">2003</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
Let us write f ðn; D; C 2kþ1 Þ for the maximal number of edges in a graph of order n and maximum degree D that contains no cycles of length 2k þ 1: For n 2 pDpn À k À 1 and n sufficiently large we show  ...  that f ðn; D; C 2kþ1 Þ ¼ Dðn À DÞ; with the unique extremal graph a complete bipartite graph. r 2002 Published by Elsevier Science (USA).  ...  Since K In=2m;Jn=2n contains no odd cycles, the maximal number of edges in a graph containing no C 2kþ1 (the extremal number for C 2kþ1 ) is In 2 =4m for sufficiently large n: The main aim of this paper  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0095-8956(02)00024-2">doi:10.1016/s0095-8956(02)00024-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lej2gf6fjngxvm4r6ljhvngl6a">fatcat:lej2gf6fjngxvm4r6ljhvngl6a</a> </span>
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Page 1784 of Mathematical Reviews Vol. , Issue 93d [page]

<span title="">1993</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Let L(G) = {2i+1| G contains a cycle of length 2i+ 1} be the set of odd cycle lengths of a graph G.  ...  J. (1-MEMP); Gydarfas, A. (H-AOS-C); Schelp, R. H. (1-MEMP) Odd cycles in graphs of given minimum degree.  ... 
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Computing independent sets in graphs with large girth

Owen J. Murphy
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
It is shown that the well-known independent set problem remains NP-complete even when restricted to graphs which contain no cycles of length less than cnk where n is the number of verticec in the graph  ...  ., Computing independent sets in graphs with large girth, Discrete Applied Mathzmatlcs 35 (1992) 167-170.  ...  S,* has cardinality no less than that of S:, and the vertices of degree 3 in S: form an independent set in G of size no less than q=q*-$np. 0 A graph is bipartite if it contains no cycles of odd length  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0166-218x(92)90041-8">doi:10.1016/0166-218x(92)90041-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/blmquubisfg25ndsw5pcqou6hm">fatcat:blmquubisfg25ndsw5pcqou6hm</a> </span>
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Graph-TSP from Steiner Cycles [chapter]

Satoru Iwata, Alantha Newman, R. Ravi
<span title="">2014</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
For a graph G, if we can find a spanning tree T and a simple cycle that contains the vertices with odd-degree in T , then we show how to combine the classic "double spanning tree" algorithm with Christofides  ...  We present an approach for the traveling salesman problem with graph metric based on Steiner cycles. A Steiner cycle is a cycle that is required to contain some specified subset of vertices.  ...  Theorem 1 For a given graph G, suppose we have a minimum spanning tree T and a simple cycle C T that contains all vertices with odd degree in T .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-12340-0_26">doi:10.1007/978-3-319-12340-0_26</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vf4loavsyvaxvjnxnemxnmnu7a">fatcat:vf4loavsyvaxvjnxnemxnmnu7a</a> </span>
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Graph-TSP from Steiner Cycles [article]

Satoru Iwata, Alantha Newman, R. Ravi
<span title="2014-07-10">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For a graph G, if we can find a spanning tree T and a simple cycle that contains the vertices with odd-degree in T, then we show how to combine the classic "double spanning tree" algorithm with Christofides  ...  We present an approach for the traveling salesman problem with graph metric based on Steiner cycles. A Steiner cycle is a cycle that is required to contain some specified subset of vertices.  ...  Theorem 1 For a given graph G, suppose we have a minimum spanning tree T and a simple cycle C T that contains all vertices with odd degree in T .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1407.2844v1">arXiv:1407.2844v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/226bop5yvreftcyqmbqoixqle4">fatcat:226bop5yvreftcyqmbqoixqle4</a> </span>
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Page 1297 of Mathematical Reviews Vol. , Issue 91C [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In this paper graphic sequences with given length and degree and having maximum [resp. minimum] weight are characterized. The weight of a simple graph G is the weight of its degree sequence.  ...  Tech. 1989, no. 5, 55-60. Suppose f(p) denotes the maximum possible number of edges in a simple graph with p vertices which contains no 3-regular subgraph.  ... 
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The Complexity of the Hamilton Cycle Problem in Hypergraphs of High Minimum Codegree

Frederik Garbe, Richard Mycroft, Marc Herbstritt
<span title="2016-02-16">2016</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/npfxqbtrtjcmndol3ijtu4shem" style="color: black;">Symposium on Theoretical Aspects of Computer Science</a> </i> &nbsp;
Given a 4-graph H, we define the total 2-pathlength of H to be the maximum sum of lengths of vertex-disjoint 2-paths in H.  ...  result is Dirac's theorem [4] , which states that any graph on n ≥ 3 vertices with minimum degree at least n 2 contains a Hamilton cycle.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.stacs.2016.38">doi:10.4230/lipics.stacs.2016.38</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/stacs/GarbeM16.html">dblp:conf/stacs/GarbeM16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p2vzgh4aqnb7vhzt5poj3y33gm">fatcat:p2vzgh4aqnb7vhzt5poj3y33gm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200318132300/https://research.birmingham.ac.uk/portal/files/25393517/stacspaper.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/36/ca/36ca5ce24081c1fa951e017788c6a3e837e0f3b2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.stacs.2016.38"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 6541 of Mathematical Reviews Vol. , Issue 93m [page]

<span title="">1993</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
This paper addresses the following extremal problem: What is the maximum number of edges in a connected graph with maximum degree D if the graph does not contain H as an induced subgraph?  ...  Summary: “The cycle length distribution of a graph of order n is (C1, C2,*+*,Cn) where c; is the number of cycles of length 7.  ... 
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Page 6817 of Mathematical Reviews Vol. , Issue 2000j [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
6817 (or, equivalently, in each 3-connected plane graph) with maximum face degree < k, there is a vertex of degree < 5 with cyclic degree < M(k).  ...  New results are also given, for example, that every 4-connected planar graph contains a TOKS, and some conjectures are offered, such as that a 4-connected graph having two disjoint odd cycles contains  ... 
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Regular Turán numbers and some Gan-Loh-Sudakov-type problems [article]

Stijn Cambie, Rémi de Joannis de Verclos, Ross J. Kang
<span title="2020-09-11">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Among other results, we prove a striking supersaturation version of Mantel's theorem in the case of a regular host graph of odd order.  ...  Motivated by a Gan-Loh-Sudakov-type problem, we introduce the regular Tur\'an numbers, a natural variation on the classical Tur\'an numbers for which the host graph is required to be regular.  ...  Let ℓ ≥ 1 and G be a non-bipartite graph with minimum degree δ > 2 n 2ℓ+1 , then G contains an odd cycle C m with m ≤ 2ℓ − 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1911.08452v2">arXiv:1911.08452v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tl4q3kapm5hevi6rm2m22rx5vu">fatcat:tl4q3kapm5hevi6rm2m22rx5vu</a> </span>
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The extremal function for cycles of length ℓ mod k [article]

Benny Sudakov, Jacques Verstraete
<span title="2016-06-28">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Burr and Erdős conjectured that for each k,ℓ∈ Z^+ such that k Z + ℓ contains even integers, there exists c_k(ℓ) such that any graph of average degree at least c_k(ℓ) contains a cycle of length ℓ mod k.  ...  Since the complete bipartite graph K_ℓ - 1,n - ℓ + 1 has no cycle of length 2ℓ mod k, it also shows c_k(ℓ) = Θ(ℓ) for ℓ = Ω( k).  ...  He would like to thank FIM for the hospitality and for creating a stimulating research environment.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1606.08532v1">arXiv:1606.08532v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/26ryzcr6afhqznf6fj2czj6mby">fatcat:26ryzcr6afhqznf6fj2czj6mby</a> </span>
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Cycle lengths in sparse graphs [article]

Benny Sudakov, Jacques Verstraete
<span title="2007-07-14">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
which gives an upper bound on the average degree of an n-vertex graph with no cycle of even length in a prescribed infinite sequence of integers.  ...  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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0707.2117v1">arXiv:0707.2117v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gved64uipnfxlnj4bdkflfdptq">fatcat:gved64uipnfxlnj4bdkflfdptq</a> </span>
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QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs [article]

Édouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, Florian Sikora
<span title="2018-02-28">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length.  ...  A (unit) disk graph is the intersection graph of closed (unit) disks in the plane.  ...  Let H be a graph with n vertices and no odd cycle shorter than δn (δ may be a function of n).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1712.05010v2">arXiv:1712.05010v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4s2pb7ozfrgtfgcdkcn627fnwe">fatcat:4s2pb7ozfrgtfgcdkcn627fnwe</a> </span>
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Approximating Maximum Edge Coloring in Multigraphs [chapter]

Uriel Feige, Eran Ofek, Udi Wieder
<span title="">2002</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We study the complexity of the following problem that we call Max edge t-coloring: given a multigraph G and a parameter t, color as many edges as possible using t colors, such that no two adjacent edges  ...  Online Max edge t-coloring The online version of Max edge t-coloring is the one in which the edges of the graph are given to the algorithm one by one.  ...  Acknowledgment This work was supported in part by a grant of the Israeli Ministry of Industry and Commerce through the consortium on Large Scale Rural Telephony.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-45753-4_11">doi:10.1007/3-540-45753-4_11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qtxggi73xretpnzjhbslia5swe">fatcat:qtxggi73xretpnzjhbslia5swe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20040612031910/http://www.wisdom.weizmann.ac.il:80/%7Eerano/Papers/MaxEdgeCol.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/aa/ba/aabaeb9424ae8574c159fa804c7189e482b85c07.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-45753-4_11"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Cycle lengths in sparse graphs

Benny Sudakov, Jacques Verstraëte
<span title="">2008</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ujm6quky2va7pmzbitmgsrksbu" style="color: black;">Combinatorica</a> </i> &nbsp;
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.  ...  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.  ...  The authors would like to thank the referee for careful reading of this manuscript.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00493-008-2300-6">doi:10.1007/s00493-008-2300-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/336a6ssplfff7pnu66yeiaabay">fatcat:336a6ssplfff7pnu66yeiaabay</a> </span>
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