A Note on Connectivity of a Cubic Graph

2021 ◽  
pp. 2142011
Author(s):  
Jinqiu Zhou ◽  
Qunfang Li ◽  
Sufang Liu
Keyword(s):  
Graph Theory ◽  
Essential Role ◽  
Cubic Graph ◽  
Edge Connectivity ◽  
Theoretical Considerations

Connectedness of a graph has a longstanding interest in combinatorial mathematics. For example, it plays an essential role in applications of graph theory and also plays a basic role in theoretical considerations. In this note, we show that the connectivity, edge connectivity, cyclically edge connectivity and essentially edge connectivity of a cubic graph are equivalent.

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 Connectivity Evaluation and Planning of a River-Lake System in East China Based on Graph Theory

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Jing Chen ◽  
Chenguang Xiao ◽  
Dan Chen
Keyword(s):  
Graph Theory ◽  
Index System ◽  
Evaluation Index System ◽  
Evaluation Index ◽  
Edge Weight ◽  
Industrial Park ◽  
Stream Network ◽  
Edge Connectivity ◽  
Edge Node ◽  
Lake System

The connectivity of the stream network plays an important role in water-mediated transport and river environments, which are threatened by the rugged development process in China. In this study, based on graph theory, a connectivity evaluation index system was built, which includes the Edge Connectivity, Edge-Node rate, Connectivity Reliability, and Edge Weight. The new evaluation standard and calculation method of each index is presented. The river-lake system of Fenhu industrial park in Jiangsu China is simplified to an Edge-Node graph and evaluated by the index system as a case study. The results indicate that the river-lake system of the research area has low Edge Connectivity, a high Edge-Node rate, and high reliability in the current connectivity level. In addition, the Edge Weight index of several channels does not satisfy the standard of the Basic Edge Weight. To solve the connectivity problems, specific project plans include broadening the unqualified channel and building canals linked with the low-connectivity lakes. The results show that, after the planning, the connectivity of the stream network in Fenhu industrial park will increase, and the connectivity evaluation index system is useful in the study area.

scholarly journals Flows in Signed Graphs with Two Negative Edges

10.37236/4458 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Edita Rollová ◽  
Michael Schubert ◽  
Eckhard Steffen
Keyword(s):  
Signed Graph ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Signed Graphs ◽  
Flow Number ◽  
Edge Connectivity ◽  
Cyclic Edge Connectivity

The presented paper studies the flow number $F(G,\sigma)$ of flow-admissible signed graphs $(G,\sigma)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\sigma)$ there is a set of cubic graphs obtained from $(G,\sigma)$ such that the flow number of $(G,\sigma)$ does not exceed the flow number of any of the cubic graphs. We prove that $F(G,\sigma) \leq 6$ if $(G,\sigma)$ contains a bridge, and $F(G,\sigma) \leq 7$ in general. We prove better bounds, if there is a cubic graph $(H,\sigma_H)$ obtained from $(G,\sigma)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,\sigma) \leq 4$ and the bound is tight. If $H$ is $3$-edge-colorable or critical or if it has a sufficient cyclic edge-connectivity, then $F(G,\sigma) \leq 6$. Furthermore, if Tutte's $5$-Flow Conjecture is true, then $(G,\sigma)$ admits a nowhere-zero $6$-flow endowed with some strong properties.

scholarly journals Cycle Lengths Modulo k in Large 3-connected Cubic Graphs, Advances in Combinatorics

2021 ◽  
Author(s):  
Kasper S. Lyngsie ◽  
Martin Merker
Keyword(s):  
Graph Theory ◽  
Minimum Degree ◽  
Average Degree ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Open Problems ◽  
The Third ◽  
Long Line ◽  
Cycle Lengths ◽  
Parity Condition

The existence of cycles with a given length is classical topic in graph theory with a plethora of open problems. Examples related to the main result of this paper include a conjecture of Burr and Erdős from 1976 asked whether for every integer $m$ and a positive odd integer $k$, there exists $d$ such that every graph with average degree at least $d$ contains a cycle of length $m$ modulo $k$; this conjecture was proven by Bollobás in [Bull. London Math. Soc. 9 (1977), 97-98]( https://doi.org/10.1112/blms/9.1.97). Another example is a problem of Erdős from the 1990s asking whether there exists $A\subseteq\mathbb{N}$ with zero density and constants $n_0$ and $d_0$ such that every graph with at least $n_0$ vertices and the average degree at least $d_0$ contains a cycle with length in the set $A$, which was resolved by Verstraete in [J. Graph Theory 49 (2005), 151-167]( https://doi.org/10.1002/jgt.20072). In 1983, Thomassen conjectured that for all integers $m$ and $k$, every graph with minimum degree $k+1$ contains a cycle of length $2m$ modulo $k$. Note that the parity condition in the first and the third conjectures is necessary because of bipartite graphs. The current paper contributes to this long line of research by proving that for every integer $m$ and a positive odd integer $k$, every sufficiently large $3$-connected cubic graph contains a cycle of length $m$ modulo $k$. The result is the best possible in the sense that the same conclusion is not true for $2$-connected cubic graphs or $3$-connected graphs with minimum degree three.

scholarly journals From bridges to cycles in spectroscopic networks

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
P. Árendás ◽  
T. Furtenbacher ◽  
A. G. Császár
Keyword(s):  
Graph Theory ◽  
High Resolution ◽  
Spectroscopic Data ◽  
Quantum States ◽  
High Resolution Spectroscopy ◽  
Edge Connectivity ◽  
Spectroscopic Measurements ◽  
New Concepts ◽  
Algorithmic Solution ◽  
Minimum Number

Abstract Spectroscopic networks provide a particularly useful representation of observed rovibronic transitions of molecules, as well as of related quantum states, whereby the states form a set of vertices connected by the measured transitions forming a set of edges. Among their several uses, SNs offer a practical framework to assess data in line-by-line spectroscopic databases. They can be utilized to help detect flawed transition entries. Methods which achieve this validation work for transitions taking part in at least one cycle in a measured spectroscopic network but they do not work for bridges. The concept of two-edge-connectivity of graph theory, introduced here to high-resolution spectroscopy, offers an elegant approach that facilitates putting the maximum number of bridges, if not all, into at least one cycle. An algorithmic solution is shown how to augment an existing spectroscopic network with a minimum number of new spectroscopic measurements selected according to well-defined guidelines. In relation to this, two metrics are introduced, ranking measurements based on their utility toward achieving the goal of two-edge-connectivity. Utility of the new concepts are demonstrated on spectroscopic data of $$^{14} {\text {NH}}_3$$ 14 NH 3 .

Constructing tensegrity structures from one-bar elementary cells

2009 ◽  
Vol 466 (2113) ◽  
pp. 45-61 ◽  
Author(s):  
Yue Li ◽  
Xi-Qiao Feng ◽  
Yan-Ping Cao ◽  
Huajian Gao
Keyword(s):  
Graph Theory ◽  
Design Strategy ◽  
The Other ◽  
Cubic Graph ◽  
Tensegrity Structures ◽  
Tensegrity Structure ◽  
Elementary Module ◽  
Shaped Cell ◽  
Selection Of

This study aimed to develop a method to construct tensegrity structures from elementary cells, defined as structures consisting of only one bar connected with a few strings. Comparison of various elementary cells leads to the further selection of the so-called ‘Z-shaped’ cell, which contains one bar and three interconnected strings, as the elementary module to assemble the Z-based spatial tensegrity structures. The graph theory is utilized to analyse the topology of strings required to construct this type of tensegrity structures. It is shown that ‘a string net can be used to construct a Z-based tensegrity structure if and only if its topology is a simple and bridgeless cubic graph’. Once the topology of strings has been determined, one can easily design the associated tensegrity structure by adding a deterministic number of bars. Two schemes are suggested for this design strategy. One is to enumerate all possible topologies of Z-based tensegrity for a specified number of bars or cells, and the other is to determine the tensegrity structure from a vertex-truncated polyhedron. The method developed in this paper allows us to construct various types of novel tensegrity structures.

scholarly journals Coverability of Graphs by Parity Regular Subgraphs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 182
Author(s):  
Mirko Petruševski ◽  
Riste Škrekovski
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Connected Graph ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Edge Connectivity ◽  
Acyclic Graph ◽  
Vertex Degrees

A graph is even (resp. odd) if all its vertex degrees are even (resp. odd). We consider edge coverings by prescribed number of even and/or odd subgraphs. In view of the 8-Flow Theorem, a graph admits a covering by three even subgraphs if and only if it is bridgeless. Coverability by three odd subgraphs has been characterized recently [Petruševski, M.; Škrekovski, R. Coverability of graph by three odd subgraphs. J. Graph Theory 2019, 92, 304–321]. It is not hard to argue that every acyclic graph can be decomposed into two odd subgraphs, which implies that every graph admits a decomposition into two odd subgraphs and one even subgraph. Here, we prove that every 3-edge-connected graph is coverable by two even subgraphs and one odd subgraph. The result is sharp in terms of edge-connectivity. We also discuss coverability by more than three parity regular subgraphs, and prove that it can be efficiently decided whether a given instance of such covering exists. Moreover, we deduce here a polynomial time algorithm which determines whether a given set of edges extends to an odd subgraph. Finally, we share some thoughts on coverability by two subgraphs and conclude with two conjectures.

Average Edge-Connectivity of Cubic Graphs

2021 ◽  
pp. 2142002
Author(s):  
Miaomiao Zhuo ◽  
Qinqin Li ◽  
Baoyindureng Wu ◽  
Xinhui An
Keyword(s):  
Disjoint Paths ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Edge Connectivity ◽  
Edge Disjoint

In this paper, we consider the concept of the average edge-connectivity [Formula: see text] of a graph [Formula: see text], defined to be the average, over all pairs of vertices, of the maximum number of edge-disjoint paths connecting these vertices. Kim and O previously proved that [Formula: see text] for any connected cubic graph on [Formula: see text] vertices. We refine their result by showing that [Formula: see text] We also characterize the graphs where equality holds.

scholarly journals Dualizing cubic graph theory

1988 ◽  
Vol 130 (1) ◽  
pp. 67-72
Author(s):  
T. McKee
Keyword(s):  
Graph Theory ◽  
Cubic Graph

scholarly journals Immersion containment and connectivity in color-critical graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Faisal N. Abu-Khzam ◽  
Michael A. Langston
Keyword(s):  
Graph Theory ◽  
Graph Coloring ◽  
Supporting Evidence ◽  
Vertex Connectivity ◽  
Edge Connectivity ◽  
Critical Graphs ◽  
International Audience ◽  
The Relationship

Graph Theory International audience The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. It is shown that a t-chromatic graph G contains either an immersed Kt or an immersed t-chromatic subgraph that is both 4-vertex-connected and t-edge-connected. This gives supporting evidence of our conjecture that if G requires at least t colors, then Kt is immersed in G.

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