scholarly journals Measures of Edge-Uncolorability of Cubic Graphs

10.37236/6848 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
M. A. Fiol ◽  
G. Mazzuoccolo ◽  
E. Steffen
Keyword(s):  
Graph Theory ◽  
Double Cover ◽  
Cubic Graphs ◽  
Open Problems ◽  
Colorable Graph ◽  
Insight Into

There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the $5$-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for $3$-edge-colorable cubic graphs, cubic graphs which are not $3$-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non $3$-edge-colorable graph is from being $3$-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such  measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.

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 A survey of graph coloring - its types, methods and applications

2012 ◽  
Vol 37 (3) ◽  
pp. 223-238 ◽  
Author(s):  
Piotr Formanowicz ◽  
Krzysztof Tanaś
Keyword(s):  
Graph Theory ◽  
Graph Coloring ◽  
Cubic Graphs ◽  
The World ◽  
Computer Scientists

Abstract Graph coloring is one of the best known, popular and extensively researched subject in the field of graph theory, having many applications and conjectures, which are still open and studied by various mathematicians and computer scientists along the world. In this paper we present a survey of graph coloring as an important subfield of graph theory, describing various methods of the coloring, and a list of problems and conjectures associated with them. Lastly, we turn our attention to cubic graphs, a class of graphs, which has been found to be very interesting to study and color. A brief review of graph coloring methods (in Polish) was given by Kubale in [32] and a more detailed one in a book by the same author. We extend this review and explore the field of graph coloring further, describing various results obtained by other authors and show some interesting applications of this field of graph theory.

scholarly journals Applications of Graph Kannan Mappings to the Damped Spring-Mass System and Deformation of an Elastic Beam

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Mudasir Younis ◽  
Deepak Singh ◽  
Adrian Petruşel
Keyword(s):  
Fixed Point ◽  
Graph Theory ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Elastic Beam ◽  
Nonlinear Problems ◽  
Point Theory ◽  
Open Problems ◽  
Mass System ◽  
Fixed Point Result

The purpose of this article is twofold. Firstly, combining concepts of graph theory and of fixed point theory, we will present a fixed point result for Kannan type mappings, in the framework of recently introduced, graphical b-metric spaces. Appropriate examples of graphs validate the established theory. Secondly, our focus is to apply the proposed results to some nonlinear problems which are meaningful in engineering and science. Some open problems are proposed.

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 Large independent sets in triangle-free cubic graphs: beyond planarity

2020 ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Gwenaël Joret ◽  
Jan Goedgebeur
Keyword(s):  
Planar Graph ◽  
Present Article ◽  
Additional Assumption ◽  
Independence Number ◽  
Independent Set ◽  
Cubic Graphs ◽  
Subcubic Graph ◽  
Color Class ◽  
Subcubic Graphs ◽  
Colorable Graph

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

Generating all the cubic graphs that have a 6-cycle double cover

2005 ◽  
Vol 19 ◽  
pp. 87-93
Author(s):  
Rodrigo S.C. Leão ◽  
Valmir C. Barbosa
Keyword(s):  
Double Cover ◽  
Cubic Graphs

scholarly journals Butterfly Triple System Algorithm Based on Graph Theory

2021 ◽  
Vol 21 (No.1) ◽  
pp. 27-49
Author(s):  
Raja'i Mohammad Aldiabat ◽  
Haslinda Ibrahim ◽  
Sharmila Karim
Keyword(s):  
Graph Theory ◽  
Special Reference ◽  
Complete Graph ◽  
Triple System ◽  
Design Theory ◽  
Combinatorial Design ◽  
Algorithm Development ◽  
Cycle System ◽  
New Type ◽  
Insight Into

In combinatorial design theory, clustering elements into a set of three elements is the heart of classifying data. This article will provide insight into formulating algorithm for a new type of triple system, called a Butterfly triple system. Basically, in this algorithm development, a starter of cyclic near-resolvable ((v-1)/2)-cycle system of the 2-fold complete graph 2K_v is employed to construct the starter of cyclic ((v-1)/2)-star decomposition of 2K_v. These starters were then decomposed into triples and classified as a starter of a cyclic Butterfly triple system. The obtained starter set generated a triple system of order A special reference for case 𝑣𝑣 ≡ 9 (mod 12) was presented to demonstrate the development of the Butterfly triple system.

scholarly journals Nikfar Domination Versus Others: Restriction, Extension Theorems and Monstrous Examples

2019 ◽  
Author(s):  
Mohammadesmail Nikfar
Keyword(s):  
Graph Theory ◽  
Real World ◽  
Dynamic State ◽  
Fuzzy Graph ◽  
Transport Planning ◽  
Open Problems ◽  
Static State ◽  
Extension Theorems ◽  
Real World Problem ◽  
Expository Article

The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between several types of domination in graphs. However, the novelty even more prominent in the newly discovered simplified presentations of several older results. Domination can be seen as arising from real-world application and extracting classical results as first described by this article.The main part of this article, concerning a new domination and older one, is presented in a narrative that answers two classical questions: (i) To what extend must closing set be dominating? (ii) How strong is the assumption of domination of a closing set? In a addition, we give an overview of the results concerning domination. The problem asks how small can a subset of vertices be and contain no edges or, more generally how can small a subset of vertices be and contain other ones. Our work was as elegant as it was unexpected being a departure from the tried and true methods of this theory that had dominated the field for one fifth a century. This expository article covers all previous definitions. The inability of previous definitions in solving even one case of real-world problems due to the lack of simultaneous attentions to the worthy both of vertices and edges causing us to make the new one. The concept of domination in a variety of graphs models such as crisp, weighted and fuzzy, has been in a spotlight. We turn our attention to sets of vertices in a fuzzy graph that are so close to all vertices, in a variety of ways, and study minimum such sets and their cardinality. A natural way to introduce and motivate our subject is to view it as a real-world problem. In its most elementary form, we consider the problem of reducing waste of time in transport planning. Our goal here is to first describe the previous definitions and the results, and then to provide an overview of the flows ideas in their articles. The final outcome of this article is twofold: (i) Solving the problem of reducing waste of time in transport planning at static state; (ii) Solving and having a gentle discussions on problem of reducing waste of time in transport planning at dynamic state. Finally, we discuss the results concerning holding domination that are independent of fuzzy graphs. We close with a list of currently open problems related to this subject. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, fuzzy graph theory and graph theory.

scholarly journals Nikfar Domination in Fuzzy Graphs

2019 ◽  
Author(s):  
Mohammadesmail Nikfar
Keyword(s):  
Graph Theory ◽  
Real World ◽  
Dynamic State ◽  
Fuzzy Graph ◽  
Transport Planning ◽  
Open Problems ◽  
Static State ◽  
Fuzzy Graphs ◽  
Real World Problem ◽  
Expository Article

The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between several types of domination in graphs. However, the novelty even more prominent in the newly discovered simplified presentations of several older results. The main part of this article, concerning a new domination and older one, is presented in a narrative that answers two classical questions: (i) To what extend must closing set be dominating? (ii) How strong is the assumption of domination of a closing set? In a addition, we give an overview of the results concerning domination. The problem asks how small can a subset of vertices be and contain no edges or, more generally how can small a subset of vertices be and contain other ones. Our work was as elegant as it was unexpected being a departure from the tried and true methods of this theory that had dominated the field for one fifth a century. This expository article covers all previous definitions. The inability of previous definitions in solving even one case of real-world problems due to the lack of simultaneous attentions to the worthy both of vertices and edges causing us to make the new one. The concept of domination in a variety of graphs models such as crisp, weighted and fuzzy, has been in a spotlight. We turn our attention to sets of vertices in a fuzzy graph that are so close to all vertices, in a variety of ways, and study minimum such sets and their cardinality. A natural way to introduce and motivate our subject is to view it as a real-world problem. In its most elementary form, we consider the problem of reducing waste of time in transport planning. Our goal here is to first describe the previous definitions and the results, and then to provide an overview of the flows ideas in their articles. The final outcome of this article is twofold: (i) Solving the problem of reducing waste of time in transport planning at static state; (ii) Solving and having a gentle discussions on problem of reducing waste of time in transport planning at dynamic state. Finally, we discuss the results concerning holding domination that are independent of fuzzy graphs. We close with a list of currently open problems related to this subject. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, fuzzy graph theory and graph theory.

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