scholarly journals Self-Avoiding Walks and the Fisher Transformation

10.37236/2659 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Geoffrey Grimmett ◽  
Zhongyang Li
Keyword(s):  
Critical Exponents ◽  
Hexagonal Lattice ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Golden Mean ◽  
Initial Graph ◽  
Connective Constant ◽  
Self Avoiding Walks

The Fisher transformation acts on cubic graphs by replacing each vertex by a triangle. We explore the action of the Fisher transformation on the set of self-avoiding walks of a cubic graph. Iteration of the transformation yields a sequence of graphs with common critical exponents, and with connective constants converging geometrically to the golden mean. We consider the application of the Fisher transformation to one of the two classes of vertices of a bipartite cubic graph. The connective constant of the ensuing graph may be expressed in terms of that of the initial graph. When applied to the hexagonal lattice, this identifies a further lattice whose connective constant may be computed rigorously.

Lower Bounds For Induced Forests in Cubic Graphs

1987 ◽  
Vol 30 (2) ◽  
pp. 193-199 ◽  
Author(s):  
J. A. Bondy ◽  
Glenn Hopkins ◽  
William Staton
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cubic Graphs ◽  
Cubic Graph

AbstractIf G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

scholarly journals Characterisation Results for Steiner Triple Systems and Their Application to Edge-Colourings of Cubic Graphs

2010 ◽  
Vol 62 (2) ◽  
pp. 355-381 ◽  
Author(s):  
Daniel Král’ ◽  
Edita Máčajov´ ◽  
Attila Pór ◽  
Jean-Sébastien Sereni
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Cubic Graphs ◽  
Steiner Triple Systems ◽  
Cubic Graph ◽  
Triple Systems ◽  
Forbidden Configurations ◽  
Edge Colourings

AbstractIt is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration C14. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations.Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph G is S-edge-colourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are S-edge-colourable for every non-projective nonaffine point-transitive Steiner triple system S.

Critical exponents of self-avoiding walks in three dimensions

1996 ◽  
Vol 53 (9) ◽  
pp. 5078-5081 ◽  
Author(s):  
N. Eizenberg ◽  
J. Klafter
Keyword(s):  
Critical Exponents ◽  
Three Dimensions ◽  
Self Avoiding Walks

scholarly journals On maximum matchings in cubic graphs with a bounded number of bridge-covering paths

1987 ◽  
Vol 36 (3) ◽  
pp. 441-447
Author(s):  
Gary Chartrand ◽  
S.F. Kapoor ◽  
Ortrud R. Oellermann ◽  
Sergio Ruiz
Keyword(s):  
Disjoint Paths ◽  
Maximum Matching ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Edge Disjoint ◽  
Bounded Number

It is proved that if G is a connected cubic graph of order p all of whose bridges lie on r edge-disjoint paths of G, then every maximum matching of G contains at least P/2 − └2r/3┘ edges. Moreover, this result is shown to be best possible.

scholarly journals Cubic identity graphs and planar graphs derived from trees

1970 ◽  
Vol 11 (2) ◽  
pp. 207-215 ◽  
Author(s):  
A. T. Balaban ◽  
Roy O. Davies ◽  
Frank Harary ◽  
Anthony Hill ◽  
Roy Westwick
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Planar Graphs ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Plane Trees ◽  
Nontrivial Identity

AbstractThe smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.

scholarly journals The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

2012 ◽  
Vol 22 (3) ◽  
pp. 765-778
Author(s):  
Piotr Formanowicz ◽  
Krzysztof Tanaś
Keyword(s):  
Graph Theory ◽  
Computer Networks ◽  
Assignment Problem ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Algorithmic Approach ◽  
Edge Colorings ◽  
Mathematical Properties

Abstract It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan–Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan–Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another possible application of the algorithms is that of being a tool for mathematicians working in the field of cubic graph theory, for discovering edge colorings with certain mathematical properties and formulating new conjectures related to the Fan–Raspaud conjecture.

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