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

1994
*
Journal of combinatorial theory. Series B (Print)
*

We also prove that for every

doi:10.1006/jctb.1994.1020
fatcat:mdc2grgk6ngjjovafcxd4nus2q
*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*...##
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On low degree k-ordered graphs

2008
*
Discrete Mathematics
*

We construct an infinite

doi:10.1016/j.disc.2007.05.024
fatcat:dbdj7njg2nadleksnxaml7p7v4
*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. ...##
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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. ...##
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Partial characterizations of clique-perfect graphs

2005
*
Electronic Notes in Discrete Mathematics
*

In this paper, we present a partial result in this direction, that is, we characterize clique-perfect

doi:10.1016/j.endm.2005.05.014
fatcat:3kfyi76ifrcxllli3c74cwfrhq
*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. ...##
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On low degree k-ordered graphs
[article]

2005
*
arXiv
*
pre-print

We construct an infinite

arXiv:math/0509411v1
fatcat:quahv66h2rh57p76rd2zjziyym
*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. ...##
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Homomorphism bounds and edge-colourings of K 4 -minor-free graphs

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

We present a necessary and sufficient condition for a

doi:10.1016/j.jctb.2017.01.001
fatcat:bklmzmxckbeg3m42vuzcdvm55q
*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*. ...##
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An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

1998
*
Combinatorica
*

We study the maximal number

doi:10.1007/pl00009804
fatcat:hmexeapyb5amnecytdslmbjmxm
*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]. ...##
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Forbidden induced pairs for perfectness and ω-colourability of graphs
[article]

2022
*
arXiv
*
pre-print

More generally, we show characterizations

arXiv:2108.07071v3
fatcat:alrkyz4advaxxjv3mkwh7urs5e
*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. ...##
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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*. ...##
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Upward-closed hereditary families in the dominance order

2022
*
Discrete Mathematics & Theoretical Computer Science
*

Motivated by the fact that threshold

doi:10.46298/dmtcs.5666
fatcat:3qv33spd2jhknirnjtc3w26dda
*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. ...##
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Polarities and 2k-cycle-free graphs

1999
*
Discrete Mathematics
*

Let C2, be the

doi:10.1016/s0012-365x(99)90107-3
fatcat:ug3cqmfrzfevlhngqdumywozq4
*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. ...##
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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 ...##
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:{unav)

2012
*
Journal of Algebraic Combinatorics
*

The proof is by an explicit construction based on the tight relationship between the spectral

doi:10.1023/a:1022453926717
fatcat:jlykk6e5grbxpphxmhbvcjwtvi
*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. ...##
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Cliques in C_4-free graphs of large minimum degree
[article]

2015
*
arXiv
*
pre-print

Hubenko, Solymosi and the first author proved (answering a question

arXiv:1509.05857v1
fatcat:6g34xas3uravlfxheazvlbynkq
*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*). ...##
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On the Turán number of some ordered even cycles

2018
*
European journal of combinatorics (Print)
*

A classical result

doi:10.1016/j.ejc.2018.05.008
fatcat:lexxmt3a2nhuppw3crxr5hppzq
*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*...
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