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Bipartite density of cubic graphs

2003
*
Discrete Mathematics
*

We ÿrst obtain the exact value for

doi:10.1016/s0012-365x(02)00490-9
fatcat:rg2l7h235fcfrdp3sgbgsdzia4
*bipartite**density**of*a*cubic*line*graph*on n vertices. ... Then we give an upper bound for the*bipartite**density**of**cubic**graphs*in terms*of*the smallest eigenvalue*of*the adjacency matrix. ... [19] to our attention, which results in a new short proof*of*Theorem 2.4. ...##
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Bipartite density of triangle-free subcubic graphs

2009
*
Discrete Applied Mathematics
*

A

doi:10.1016/j.dam.2008.07.007
fatcat:d43pemkbf5b23crqewmkq7t2c4
*graph*is subcubic if its maximum degree is at most 3. The*bipartite**density**of*a*graph*G is defined as b(G) = max{|E(B)|/|E(G)| : B is a*bipartite*subgraph*of*G}. ... It was conjectured by Bondy and Locke that if G is a triangle-free subcubic*graph*, then b(G) ≥ 4 5 and equality holds only if G is in a list*of*seven small*graphs*. ... They further proved that the Petersen*graph*and the dodecahedron (see Fig. 1 ) are the only triangle-free*cubic**graphs*with*bipartite**density*4 5 . ...##
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Disjoint Triangles of a Cubic Line Graph

2004
*
Graphs and Combinatorics
*

Moreover, the equitable chromatic number, acyclic chromatic number and

doi:10.1007/s00373-004-0551-6
fatcat:iwostnemgfdcjhe35jb7qv4j2u
*bipartite**density**of*G are 3, 3, 7 9 respectively. ... In this paper, we prove that a*cubic*line*graph*G on n vertices rather than the complete*graph*K 4 has b n 3 c vertex-disjoint triangles and the vertex independence number b n 3 c. ... The authors are grateful to the referees for valuable suggestions and comments which results in a great improvement*of*the original manuscript. ...##
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On a bipartition problem of Bollobás and Scott

2009
*
Combinatorica
*

The

doi:10.1007/s00493-009-2381-x
fatcat:icalh57sdbdubcrxx2xtildkgi
*bipartite**density**of*a*graph*G is max{|E(H)|/|E(G)| : H is a*bipartite*subgraph*of*G}. It is NP-hard to determine the*bipartite**density**of*any triangle-free*cubic**graph*. ... Let H denote the collection*of*all connected*cubic**graphs*which have*bipartite**density*4 5 and contain biased maximum*bipartite*subgraphs. Bollobás and Scott asked which*cubic**graphs*belong to H. ...*bipartite**density*4 5 . ...##
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Dense Graph Partitioning on sparse and dense graphs
[article]

2022
*
arXiv
*
pre-print

In particular, we prove that it is NP-hard on dense

arXiv:2107.13282v3
fatcat:l2eaob2omzfizh3sdvkykrj5yy
*bipartite**graphs*as well as on*cubic**graphs*. ... We consider the problem*of*partitioning a*graph*into a non-fixed number*of*non-overlapping subgraphs*of*maximum*density*. ... The*density*d(G n,m )*of*a complete*bipartite**graph*G n,m is greater than or equal to the*density*d(P)*of*any partition P*of*G n,m . Proof. ...##
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Page 8942 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

Summary: “We first obtain the exact value for

*bipartite**density**of*a*cubic*line*graph*on n vertices. ... | (IL-TECH; Haifa)*Bipartite**density**of**cubic**graphs*. (English summary) Discrete Math. 260 (2003), no. 1-3, 27-35. ...##
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LDPC codes constructed from cubic symmetric graphs
[article]

2020
*
arXiv
*
pre-print

Information on the LDPC codes constructed from

arXiv:2002.06690v1
fatcat:7u2mq4vcqfalxm7jd2rr3ilbii
*bipartite**cubic*symmetric*graphs*with less than 200 vertices is presented as well. ... In this paper we present a construction*of*LDPC codes from*cubic*symmetric*graphs*. ... In this paper we construct LDPC codes using*bipartite**cubic*symmetric*graphs*as Tanner*graphs**of*LDPC codes. ...##
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Page 6549 of Mathematical Reviews Vol. , Issue 93m
[page]

1993
*
Mathematical Reviews
*

Given vertex-disjoint

*graphs*G,; and G>*of*order n, and given a k-regular*bipartite**graph*B having the two sets*of*vertices for its*bipartition*, we call G; U G2UB a generalized prism A;(G, G2) over G; ... Summary: “The*density**of*a nontrivial*graph*G is defined by g(G) =|E(G)|/(V(G) — @(G)), where w@(G) denotes the number*of*components*of*G. ...##
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Page 5669 of Mathematical Reviews Vol. , Issue 88k
[page]

1988
*
Mathematical Reviews
*

Several easy variations

*of*the general problem are considered. In particular, the paper supplies an O(n + m) algorithm to find a minimum m*density*matching*of*size n*of*any complete*bipartite**graph*. ... On*bipartite*matchings*of*minimum*density*. J. Algorithms 8 (1987), no. 4, 480-502. Suppose two parallel horizontal lines are given, r and s, say. Also, suppose P [resp. ...##
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Page 1724 of Mathematical Reviews Vol. , Issue 87d
[page]

1987
*
Mathematical Reviews
*

Suppose z denotes the independence ratio

*of*a*graph*G, the pro- portion*of*vertices in a largest independent set in G, and b denotes the*bipartite**density*, the proportion*of*edges in a largest*bipartite*... The result is potentially useful for deducing properties*of*the chromatic polynomial*of*a*graph*. E. J. Farrell (Trinidad) Locke, Stephen C. (1-FLAT)*Bipartite**density*and the independence ratio. J. ...##
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Eigenvalue spacings for regular graphs
[article]

2003
*
arXiv
*
pre-print

We carry out a numerical study

arXiv:hep-th/0310002v1
fatcat:qkags2qkvzd2zhkifndxctcu6i
*of*fluctuations in the spectrum*of*regular*graphs*. ... A review*of*the basic facts on*graphs*and their spectra is included. ... (a) a*cubic**graph*on 2000 vertices, (b) a 5-valent*graph*on 500 vertices. Figure 2 . 2 Level spacing distribution*of*a*cubic**graph*on 2000 vertices against GOE. ...##
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Triangle-free subcubic graphs with minimum bipartite density

2008
*
Journal of combinatorial theory. Series B (Print)
*

It is an NP-hard problem to determine the

doi:10.1016/j.jctb.2007.09.001
fatcat:5dmqnvab7rcy3plq6lniqrqhwu
*bipartite**density**of*any given trianglefree*cubic**graph*. ... dodecahedron are the only triangle-free*cubic**graphs*with*bipartite**density*4 5 . ... dodecahedron (shown in Fig. 1 ) are the only*cubic**graphs*with*bipartite**density*4 5 ...##
###
Eigenvalue Spacings for Regular Graphs
[chapter]

1999
*
IMA Volumes in Mathematics and its Applications
*

We carry out a numerical study

doi:10.1007/978-1-4612-1544-8_12
fatcat:37rbwzhxevhtbdfpaz4bvi6op4
*of*fluctuations in the spectrum*of*regular*graphs*. ... A review*of*the basic facts on*graphs*and their spectra is included. ... Figure 4 . 4 Distribution*of*normalized discrepancies for*cubic**graphs*vs. ...##
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Page 9585 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

Mohan (New Delhi)
05C

*Graph*theory 2004m:05172 2004m:05169 05C50 Stevanovic, Dragan (YU-NISSM; Nis)*Bipartite**density**of**cubic**graphs*: the case*of*equality. ... Zhang [Discrete Math. 260 (2003), no. 1-3, 27-35; MR 2003m:05096] obtained an upper bound for the*bipartite**density**of**cubic**graphs*in terms*of*the smallest eigenvalue*of*an adjacency matrix and attempted ...##
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Page 3520 of Mathematical Reviews Vol. , Issue 84i
[page]

1984
*
Mathematical Reviews
*

Ellingham [“Non-Hamiltonian 3-connected

*cubic**bipartite**graphs*”, Research Report No. 28, Dept.*of*Math., Univ. ...*Bipartite**cubic**graphs*and a shortness exponent. Discrete Math. 44 (1983), no. 3, 327-330. The author constructs a non-Hamiltonian*bipartite*3-connected*graph*with only 78 vertices. ...
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