scholarly journals Some Results on the Structure of Multipoles in the Study of Snarks

10.37236/3629 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
M. A. Fiol ◽  
J. Vilaltella
Keyword(s):  
Lower Bound ◽  
Edge Coloring ◽  
The State ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Order ◽  
Exact Number ◽  
Empty Subset ◽  
Free Edges

Multipoles are the pieces we obtain by cutting some edges of a cubic graph in one or more points. As a result of the cut, a multipole $M$ has vertices attached to a dangling edge with one free end, and isolated edges with two free ends. We refer to such free ends as semiedges, and to isolated edges as free edges. Every 3-edge-coloring of a multipole induces a coloring or state of its semiedges, which satisfies the Parity Lemma. Multipoles have been extensively used in the study of snarks, that is, cubic graphs which are not 3-edge-colorable. Some results on the states and structure of the so-called color complete and color closed multipoles are presented. In particular, we give lower and upper linear bounds on the minimum order of a color complete multipole, and compute its exact number of states. Given two multipoles $M_1$ and $M_2$ with the same number of semiedges, we say that $M_1$ is reducible to $M_2$ if the state set of $M_2$ is a non-empty subset of the state set of $M_1$ and $M_2$ has less vertices than $M_1$. The function $v(m)$ is defined as the maximum number of vertices of an irreducible multipole with $m$ semiedges. The exact values of  $v(m)$ are only known for $m\le 5$. We prove that tree and cycle multipoles are irreducible and, as a byproduct, that $v(m)$ has a linear lower bound.

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 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 Constructions for Cubic Graphs with Large Girth

10.37236/1386 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Norman Biggs
Keyword(s):  
Lower Bound ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimal Graph ◽  
Minimum Value ◽  
Open Questions ◽  
G 12 ◽  
Large Girth ◽  
Selection Of ◽  
Finite Constant

The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer $\mu_0(g)$, the smallest number of vertices for which a cubic graph with girth at least $g$ exists, and furthermore, the minimum value $\mu_0(g)$ is attained by a graph whose girth is exactly $g$. The values of $\mu_0(g)$ when $3 \le g \le 8$ have been known for over thirty years. For these values of $g$ each minimal graph is unique and, apart from the case $g=7$, a simple lower bound is attained. This paper is mainly concerned with what happens when $g \ge 9$, where the situation is quite different. Here it is known that the simple lower bound is attained if and only if $g=12$. A number of techniques are described, with emphasis on the construction of families of graphs $\{ G_i\}$ for which the number of vertices $n_i$ and the girth $g_i$ are such that $n_i\le 2^{cg_i}$ for some finite constant $c$. The optimum value of $c$ is known to lie between $0.5$ and $0.75$. At the end of the paper there is a selection of open questions, several of them containing suggestions which might lead to improvements in the known results. There are also some historical notes on the current-best graphs for girth up to 36.

scholarly journals Exponential Domination in Subcubic Graphs

10.37236/5711 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Stéphane Bessy ◽  
Pascal Ochem ◽  
Dieter Rautenbach
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Discrete Math ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Order ◽  
Natural Variant ◽  
Dominating Set Problem ◽  
Subcubic Graphs ◽  
Domination In Graphs

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $\gamma_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$.In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If $G$ is a connected subcubic graph of order $n(G)$, then $$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq \gamma_e(G)\leq \frac{1}{3}(n(G)+2).$$ For every $\epsilon>0$, there is some $g$ such that $\gamma_e(G)\leq \epsilon n(G)$ for every cubic graph $G$ of girth at least $g$. For every $0<\alpha<\frac{2}{3\ln(2)}$, there are infinitely many cubic graphs $G$ with $\gamma_e(G)\leq \frac{3n(G)}{\ln(n(G))^{\alpha}}$. If $T$ is a subcubic tree, then $\gamma_e(T)\geq \frac{1}{6}(n(T)+2).$ For a given subcubic tree, $\gamma_e(T)$ can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs.

scholarly journals Smallest cubic and quartic graphs with a given number of cutpoints and bridges

1982 ◽  
Vol 5 (1) ◽  
pp. 41-48 ◽  
Author(s):  
Gary Chartrand ◽  
Farrokh Saba ◽  
John K. Cooper ◽  
Frank Harary ◽  
Curtiss E. Wall
Keyword(s):  
Positive Integer ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Order ◽  
Positive Integers

For positive integersbandc, withceven, satisfying the inequalitiesb+1≤c≤2b, the minimum order of a connected cubic graph withbbridges andccutpoints is computed. Furthermore, the structure of all such smallest cubic graphs is determined. For each positive integerc, the minimum order of a quartic graph withccutpoints is calculated. Moreover, the structure and number of all such smallest quartic graphs are determined.

scholarly journals Density Guarantee on Finding Multiple Subgraphs and Subtensors

2021 ◽  
Vol 15 (5) ◽  
pp. 1-32
Author(s):  
Quang-huy Duong ◽  
Heri Ramampiaro ◽  
Kjetil Nørvåg ◽  
Thu-lan Dam
Keyword(s):  
Lower Bound ◽  
State Of The Art ◽  
The State ◽  
The Other ◽  
Exact Methods ◽  
Practical Solution ◽  
Novel Approach ◽  
Wide Range ◽  
Real World Datasets ◽  
Tensor Data

Dense subregion (subgraph & subtensor) detection is a well-studied area, with a wide range of applications, and numerous efficient approaches and algorithms have been proposed. Approximation approaches are commonly used for detecting dense subregions due to the complexity of the exact methods. Existing algorithms are generally efficient for dense subtensor and subgraph detection, and can perform well in many applications. However, most of the existing works utilize the state-or-the-art greedy 2-approximation algorithm to capably provide solutions with a loose theoretical density guarantee. The main drawback of most of these algorithms is that they can estimate only one subtensor, or subgraph, at a time, with a low guarantee on its density. While some methods can, on the other hand, estimate multiple subtensors, they can give a guarantee on the density with respect to the input tensor for the first estimated subsensor only. We address these drawbacks by providing both theoretical and practical solution for estimating multiple dense subtensors in tensor data and giving a higher lower bound of the density. In particular, we guarantee and prove a higher bound of the lower-bound density of the estimated subgraph and subtensors. We also propose a novel approach to show that there are multiple dense subtensors with a guarantee on its density that is greater than the lower bound used in the state-of-the-art algorithms. We evaluate our approach with extensive experiments on several real-world datasets, which demonstrates its efficiency and feasibility.

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 State Complexity of Neighbourhoods and Approximate Pattern Matching

2018 ◽  
Vol 29 (02) ◽  
pp. 315-329 ◽  
Author(s):  
Timothy Ng ◽  
David Rappaport ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Pattern Matching ◽  
Finite Automaton ◽  
The State ◽  
Worst Case ◽  
State Complexity ◽  
Approximate Pattern Matching ◽  
The Given ◽  
Nondeterministic Finite Automaton ◽  
Distance Formula

The neighbourhood of a language [Formula: see text] with respect to an additive distance consists of all strings that have distance at most the given radius from some string of [Formula: see text]. We show that the worst case deterministic state complexity of a radius [Formula: see text] neighbourhood of a language recognized by an [Formula: see text] state nondeterministic finite automaton [Formula: see text] is [Formula: see text]. In the case where [Formula: see text] is deterministic we get the same lower bound for the state complexity of the neighbourhood if we use an additive quasi-distance. The lower bound constructions use an alphabet of size linear in [Formula: see text]. We show that the worst case state complexity of the set of strings that contain a substring within distance [Formula: see text] from a string recognized by [Formula: see text] is [Formula: see text].

Utilitarianism

2019 ◽  
pp. 9-21
Author(s):  
Torbjörn Tännsjö
Keyword(s):  
Distributive Justice ◽  
Crucial Role ◽  
Moral Theory ◽  
The State ◽  
Point Of View ◽  
Subjective Time ◽  
Worth Living ◽  
Common Currency ◽  
Exact Number ◽  
Single Scale

Utilitarianism is the idea that we ought to maximize the sum total of happiness. The notion of happiness is clarified. Happiness is taken in a subjective and empirical sense, as a kind of mood. Affirmative answers to the following questions are provided: What is happiness? Can it be measured? Can we compare it between persons? Can it function as a common currency when the different theories of distributive justice are compared? What about the heterogeneity objection? Can very different kinds of happiness be measured on a single scale? In the answers to these questions the idea of a least noticeable difference with respect to happiness plays a crucial role. It is conjectured that, if a person is in a certain mood (momentarily), then there exists an exact number of just noticeable changes for the worse or the better to the point where life is just worth living. Many different conditions can contribute to cause a person to be at the state where she is. A distinction of the utmost importance between physical and subjective time is introduced and a claim is made that what matters, from the point of view of moral theory, is subjective time.

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.

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