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Properties of Certain Families of 2k-Cycle-Free Graphs

F Lazebnik, V.A Ustimenko, A.J Woldar
1994 Journal of combinatorial theory. Series B (Print)  
We also prove that for every family of 2k-cycle free extremal graphs (i.e. graphs having the greatest size among all 2k-cycle free graphs of the same order), all but finitely many such graphs must be either  ...  For any such family G whose members are bipartite and of girth at least 2k + 2, and every integer t, 2 ≤ t ≤ k − 1, we construct a family G t of graphs of same magnitude r, of constant greater than λ,  ...  Then, for any t, 2 ≤ t ≤ k − 1, there exists a family G t of 2k-cycle free graphs with magnitude r and constant λ ≥ t( 2 t+1 ) r λ > λ, all of whose members are bipartite and contain each of the cycles  ... 
doi:10.1006/jctb.1994.1020 fatcat:mdc2grgk6ngjjovafcxd4nus2q

On low degree k-ordered graphs

Karola Mészáros
2008 Discrete Mathematics  
We construct an infinite family of graphs, which we call bracelet graphs, that are (k − 1)-regular and are k-ordered hamiltonian for odd k.  ...  A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v 1 , . . . , v k of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing  ...  Chebikin, On k-edge-ordered graphs, Discrete Math. 281 (2004) 115-128.  ... 
doi:10.1016/j.disc.2007.05.024 fatcat:dbdj7njg2nadleksnxaml7p7v4

Page 38 of Mathematical Reviews Vol. , Issue 95a [page]

1995 Mathematical Reviews  
J. (1-VLNV; Villanova, PA) Properties of certain families of 2k-cycle-free graphs. (English summary) J. Combin. Theory Ser. B 60 (1994), no. 2, 293-298.  ...  Let k > 3, and let ¥ be a family of 2k-cycle-free graphs with magnitude r > 1 and constant A > 0, the members of which are bipartite graphs of girth at least 2k + 2.  ... 

Partial characterizations of clique-perfect graphs

Flavia Bonomo, Maria Chudnovsky, Guillermo Durán
2005 Electronic Notes in Discrete Mathematics  
In this paper, we present a partial result in this direction, that is, we characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to a certain class  ...  The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known.  ...  A family of sets S is said to satisfy the Helly property if every subfamily of it, consisting of pairwise intersecting sets, has a common element.  ... 
doi:10.1016/j.endm.2005.05.014 fatcat:3kfyi76ifrcxllli3c74cwfrhq

On low degree k-ordered graphs [article]

Karola Meszaros
2005 arXiv   pre-print
We construct an infinite family of graphs, which we call bracelet graphs, that are (k-1)-regular and are k-ordered hamiltonian for odd k.  ...  A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v_1, ..., v_k of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing  ...  This research was performed at the University of Minnesota Duluth under the supervision of Professor Joseph A. Gallian.  ... 
arXiv:math/0509411v1 fatcat:quahv66h2rh57p76rd2zjziyym

Homomorphism bounds and edge-colourings of K 4 -minor-free graphs

Laurent Beaudou, Florent Foucaud, Reza Naserasr
2017 Journal of combinatorial theory. Series B (Print)  
We present a necessary and sufficient condition for a graph of odd-girth 2k+1 to bound the class of K_4-minor-free graphs of odd-girth (at least) 2k+1, that is, to admit a homomorphism from any such K_  ...  Using this condition, we first prove that every K_4-minor free graph of odd-girth 2k+1 admits a homomorphism to the projective hypercube of dimension 2k.  ...  These conditions are given in terms of the existence of a certain weighted graph (that we call a k-partial distance graph of B) containing B as a subgraph, and that satisfies certain properties.  ... 
doi:10.1016/j.jctb.2017.01.001 fatcat:bklmzmxckbeg3m42vuzcdvm55q

An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

Y. Kohayakawa, B. Kreuter, A. Steger
1998 Combinatorica  
We study the maximal number of edges a C2k-free subgraph of a random graph Gn;p may have, obtaining best possible results for a range of p = p(n).  ...  Two main tools are used here: the rst one is an upper bound for the number of graphs with large even-girth, i.e., graphs without short even cycles, with a given number of vertices and edges, and satisfying  ...  F uredi for informing us about the results of D.J. Kleitman and D.B. Wilson, and to these authors for letting us have a preliminary copy of 15].  ... 
doi:10.1007/pl00009804 fatcat:hmexeapyb5amnecytdslmbjmxm

Forbidden induced pairs for perfectness and ω-colourability of graphs [article]

Maria Chudnovsky, Adam Kabela, Binlong Li, Petr Vrána
2022 arXiv   pre-print
More generally, we show characterizations of pairs { X, Y } for perfectness and ω-colourability of all connected { X, Y }-free graphs which are of independence at least 3, distinct from an odd cycle, and  ...  We characterise the pairs of graphs { X, Y } such that all { X, Y }-free graphs (distinct from C_5) are perfect.  ...  The work of the second and fourth author was supported by projects 17-04611S and 20-09525S of the Czech Science Foundation.  ... 
arXiv:2108.07071v3 fatcat:alrkyz4advaxxjv3mkwh7urs5e

Page 6611 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
This article examines the homomorphic properties of certain subgraphs of the unit distance graph D.  ...  Denote by /(t) the set of natural numbers i, |¢/2| <i <t—2, with the property that in every connected P,-free graph G, the i-center C;(G) is a connected graph.  ... 

Upward-closed hereditary families in the dominance order

Michael D. Barrus, Jean A. Guillaume
2022 Discrete Mathematics & Theoretical Computer Science  
Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever  ...  The majorization relation orders the degree sequences of simple graphs into posets called dominance orders.  ...  Acknowledgments The authors wish to thank the anonymous referees for their careful reading of the manuscript and their helpful suggestions.  ... 
doi:10.46298/dmtcs.5666 fatcat:3qv33spd2jhknirnjtc3w26dda

Polarities and 2k-cycle-free graphs

Felix Lazebnik, Vasiliy A. Ustimenko, Andrew J. Woldar
1999 Discrete Mathematics  
Let C2, be the cycle on 2k vertices, and let ex(v, C2k) denote the greatest number of edges in a simple graph on v vertices which contains no subgraph isomorphic to C2,.  ...  We are grateful to an anonymous referee for several corrections, several references and many suggestions which improved the original version of this paper.  ...  For other values of k the existence of a 2k-cycle-free graph with O(vl+~ -) edges has not been established.  ... 
doi:10.1016/s0012-365x(99)90107-3 fatcat:ug3cqmfrzfevlhngqdumywozq4

Page 4666 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
Bondy, asserts that a 2-edge-connected graph of order n has a family of at most n—1 cycles so that every edge of the graph lies in precisely two members of the family.  ...  Combin. 16 (1995), no. 1, 17-21; MR 95m:05195] conjectured that a minimally even pair free graph is either an odd cycle of length at least five, the complement of an even or odd cycle of length at least  ... 


Noga Alon
2012 Journal of Algebraic Combinatorics  
The proof is by an explicit construction based on the tight relationship between the spectral properties of a regular graph and its expansion properties.  ...  A similar technique provides a simple construction of triangle-free graphs with independence number m on Ω(m 4/3 ) vertices, improving previously known explicit constructions by Erdős and by Chung, Cleve  ...  The problem of finding an accurate estimate remains open. Acknowledgment I would like to thank Jan van den Heuvel for helpful discussions.  ... 
doi:10.1023/a:1022453926717 fatcat:jlykk6e5grbxpphxmhbvcjwtvi

Cliques in C_4-free graphs of large minimum degree [article]

A. Gyarfas, G. N. Sarkozy
2015 arXiv   pre-print
Hubenko, Solymosi and the first author proved (answering a question of Erdős) a peculiar property of C_4-free graphs: C_4 graphs with n vertices and average degree at least cn contain a complete subgraph  ...  Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2k-regular C_4-free graphs on 4k+1 vertices contain a clique of size k+1.  ...  The class of C 4 -free graphs have been studied from many points of view, for example they appear in the theory of perfect graphs (as families containing chordal graphs).  ... 
arXiv:1509.05857v1 fatcat:6g34xas3uravlfxheazvlbynkq

On the Turán number of some ordered even cycles

Ervin Győri, Dániel Korándi, Abhishek Methuku, István Tomon, Casey Tompkins, Máté Vizer
2018 European journal of combinatorics (Print)  
A classical result of Bondy and Simonovits in extremal graph theory states that if a graph on n vertices contains no cycle of length 2k then it has at most O(n^1+1/k) edges.  ...  In particular, we show that the maximum number of edges in an ordered graph avoiding bordered cycles of length at most 2k is Θ(n^1+1/k).  ...  After our paper was submitted, we learned that Timmons [19] also studied the Turán number of ordered cycles, and observed that ex < (n, C 2k ) = O(n 1+1/k ) for the family C 2k of all ordered 2k-cycles  ... 
doi:10.1016/j.ejc.2018.05.008 fatcat:lexxmt3a2nhuppw3crxr5hppzq
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