scholarly journals On the genera of polyhedral embeddings of cubic graph

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Gunnar Brinkmann ◽  
Thomas Tucker ◽  
Nico Van Cleemput
Keyword(s):  
Efficient Algorithm ◽  
Computational Results ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
High Genus ◽  
Theoretical Maximum ◽  
Polyhedral Embedding

In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.

scholarly journals Constructing families of cospectral regular graphs

2020 ◽  
Vol 29 (5) ◽  
pp. 664-671
Author(s):  
M. Haythorpe ◽  
A. Newcombe
Keyword(s):  
Characteristic Polynomial ◽  
Regular Graphs ◽  
Computational Results ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Simple Method ◽  
Special Cases ◽  
Adjacency Matrices

AbstractA set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid for special cases of a property introduced by Schwenk. For the case of cubic (3-regular) graphs, computational results are given which show that the construction generates a large proportion of the cubic graphs, which are cospectral with another cubic graph.

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.

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.

SIMPLE YET EFFICIENT ALGORITHM FOR SOLVING NONLINEAR EQUATIONS

2010 ◽  
Vol 01 (04) ◽  
pp. 509-522
Author(s):  
SANJAY KUMAR KHATTRI ◽  
MUHAMMAD ASLAM NOOR
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Newton Method ◽  
Efficient Algorithm ◽  
Iterative Process ◽  
Computational Results ◽  
Numerical Work ◽  
Software Packages ◽  
Practical Algorithm ◽  
Convergence Orders

In this work, we develop a simple yet robust and highly practical algorithm for constructing iterative methods of higher convergence orders. The algorithm can be easily implemented in software packages for achieving desired convergence orders. Convergence analysis shows that the algorithm can develop methods of various convergence orders which is also supported through the numerical work. The algorithm is shown to converge even if the derivative of the function vanishes during the iterative process. Computational results ascertain that the developed algorithm is efficient and demonstrate equal or better performance as compared with other well known methods and the classical Newton method.

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.

scholarly journals Snarks with total chromatic number 5

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Gunnar Brinkmann ◽  
Myriam Preissmann ◽  
Diana Sasaki
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Total Coloring ◽  
Computer Search ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Total Chromatic Number ◽  
Vertex Number ◽  
Chordless Cycle

Graph Theory International audience A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by χT(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with χT = 4 are said to be Type 1, and cubic graphs with χT = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n≥40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

Contractible Edges in 3-Connected Cubic Graphs

2021 ◽  
pp. 2150014
Author(s):  
Shuai Kou ◽  
Chengfu Qin ◽  
Weihua Yang
Keyword(s):  
Connected Graph ◽  
Independent Set ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Better Than

An edge [Formula: see text] in a 3-connected graph [Formula: see text] is contractible if the contraction [Formula: see text] is still [Formula: see text]-connected. Let [Formula: see text] be the set of contractible edges of [Formula: see text], [Formula: see text] be the set of vertices adjacent to three vertices of a triangle △. It has been proved that [Formula: see text] in a 3-connected graph [Formula: see text] of order at least 5. In this note [Formula: see text] is a 3-connected cubic graph containing [Formula: see text] triangles, at least [Formula: see text] vertices and with every [Formula: see text] an independent set. Then [Formula: see text]. This is a bound better than [Formula: see text] under some conditions.

Sign in / Sign up

Export Citation Format

Share Document