scholarly journals On a Conjecture Concerning the Petersen Graph

10.37236/507 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Donald Nelson ◽  
Michael D. Plummer ◽  
Neil Robertson ◽  
Xiaoya Zha
Keyword(s):  
Connected Graph ◽  
Independent Set ◽  
Petersen Graph ◽  
Original Form ◽  
Odd Cycle ◽  
Special Case

Robertson has conjectured that the only 3-connected internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We prove this conjecture in the special case where the graphs involved are also cubic. Moreover, this proof does not require the internal-4-connectivity assumption. An example is then presented to show that the assumption of internal 4-connectivity cannot be dropped as an hypothesis in the original conjecture. We then summarize our results aimed toward the solution of the conjecture in its original form. In particular, let $G$ be any 3-connected internally-4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord. If $C$ is any girth cycle in $G$ then $N(C)\backslash V(C)$ cannot be edgeless, and if $N(C) \backslash V(C)$ contains a path of length at least 2, then the conjecture is true. Consequently, if the conjecture is false and $H$ is a counterexample, then for any girth cycle $C$ in $H$, $N(C) \backslash V(C)$ induces a nontrivial matching $M$ together with an independent set of vertices. Moreover, $M$ can be partitioned into (at most) two disjoint non-empty sets where we can precisely describe how these sets are attached to cycle $C$.

scholarly journals On a Conjecture Concerning the Petersen Graph: Part II

10.37236/2315 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Plummer ◽  
Xiaoya Zha
Keyword(s):  
Connected Graph ◽  
Petersen Graph ◽  
Odd Cycle

Robertson conjectured that the only 3-connected, internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph.  We provide a counterexample to this conjecture.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

Kruskal's theorem for matroids

1968 ◽  
Vol 64 (1) ◽  
pp. 3-4 ◽  
Author(s):  
D. J. A. Welsh
Keyword(s):  
Vector Space ◽  
Finite Subset ◽  
Spanning Tree ◽  
General Theorem ◽  
Independent Set ◽  
Easy Method ◽  
Maximal Weight ◽  
Linearly Independent ◽  
Kruskal’S Theorem ◽  
Special Case

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

The Homology of Abelian Covers of Knotted Graphs

1999 ◽  
Vol 51 (5) ◽  
pp. 1035-1072
Author(s):  
R. A. Litherland
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Petersen Graph ◽  
Infinite Class ◽  
Colored Graphs ◽  
Branched Cover ◽  
Trivalent Graph ◽  
Abelian Covers ◽  
Index 2

AbstractLet be a regular branched cover of a homology 3-sphere M with deck group and branch set a trivalent graph Γ; such a cover is determined by a coloring of the edges of Γ with elements of G. For each index-2 subgroup H of G, MH = /H is a double branched cover of M. Sakuma has proved that H1() is isomorphic, modulo 2-torsion, to ⊕HH1(MH), and has shown that H1() is determined up to isomorphism by ⊕HH1(MH) in certain cases; specifically, when d = 2 and the coloring is such that the branch set of each cover MH → M is connected, and when d = 3 and Γ is the complete graph K4. We prove this for a larger class of coverings: when d = 2, for any coloring of a connected graph; when d = 3 or 4, for an infinite class of colored graphs; and when d = 5, for a single coloring of the Petersen graph.

scholarly journals Budget Feasible Mechanisms on Matroids

Algorithmica ◽  
2020 ◽  
Author(s):  
Stefano Leonardi ◽  
Gianpiero Monaco ◽  
Piotr Sankowski ◽  
Qiang Zhang
Keyword(s):  
Polynomial Time ◽  
Private Information ◽  
Independent Set ◽  
Deterministic Algorithm ◽  
Matroid Intersection ◽  
Practical Applications ◽  
Incentive Compatible ◽  
The Cost ◽  
Special Case ◽  
Budget Feasible

AbstractMotivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) . Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive valuations over the elements of E. The goal is to design an incentive compatible budget feasible mechanism which procures an independent set of the matroid of largest possible value. We also consider the more general case of the pair $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) satisfying only the hereditary property. This includes matroids as well as matroid intersection. We show that, given a polynomial time deterministic algorithm that returns an $$\alpha $$ α -approximation to the problem of finding a maximum-value independent set in $${\mathcal {M}}$$ M , there exists an individually rational, truthful and budget feasible mechanism which is $$(3\alpha +1)$$ ( 3 α + 1 ) -approximated and runs in polynomial time, thus yielding also a 4-approximation for the special case of matroids.

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

Quadratic-monomial generated domains from mixed signed, directed graphs

2019 ◽  
Vol 29 (02) ◽  
pp. 279-308
Author(s):  
Michael A. Burr ◽  
Drew J. Lipman
Keyword(s):  
Directed Graphs ◽  
Complete Characterization ◽  
Combinatorial Structure ◽  
Signed Directed Graph ◽  
Odd Cycle ◽  
Combinatorial Classification ◽  
Special Case ◽  
Text Condition

Determining whether an arbitrary subring [Formula: see text] of [Formula: see text] is a normal or Cohen-Macaulay domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. We provide a complete characterization of the normality, normalizations, and Serre’s [Formula: see text] condition for quadratic-monomial generated domains. For a quadratic-monomial generated domain [Formula: see text], we develop a combinatorial structure that assigns, to each quadratic monomial of the ring, an edge in a mixed signed, directed graph [Formula: see text], i.e. a graph with signed edges and directed edges. We classify the normality and the normalizations of such rings in terms of a generalization of the combinatorial odd cycle condition on [Formula: see text]. We also generalize and simplify a combinatorial classification of Serre’s [Formula: see text] condition for such rings and construct non-Cohen–Macaulay rings.

scholarly journals The K -Size Edge Metric Dimension of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Tanveer Iqbal ◽  
Muhammad Naeem Azhar ◽  
Syed Ahtsham Ul Haq Bokhary
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph

In this paper, a new concept k -size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.

scholarly journals On the maximum orders of an induced forest, an induced tree, and a stable set

2014 ◽  
Vol 24 (2) ◽  
pp. 199-215
Author(s):  
Alain Hertz ◽  
Odile Marcotte ◽  
David Schindl
Keyword(s):  
Connected Graph ◽  
Stable Set ◽  
Stability Number ◽  
Maximum Order ◽  
The Stability ◽  
Special Case

Let G be a connected graph, n the order of G, and f (resp. t) the maximum order of an induced forest (resp. tree) in G. We show that f - t is at most n - ?2?n-1?. In the special case where n is of the form a2 + 1 for some even integer a ? 4, f - t is at most n - ?2?n-1?-1. We also prove that these bounds are tight. In addition, letting ? denote the stability number of G, we show that ? - t is at most n + 1- ?2?2n? this bound is also tight.

Some Remarks on Ramsay's Theorem

1964 ◽  
Vol 7 (4) ◽  
pp. 619-622 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Independent Set ◽  
Infinite Subset ◽  
Infinite Graph ◽  
Complete Subgraph ◽  
Special Case

A special case of a well known theorem of Ramsay [3] states that an infinite graph either contains an infinite complete subgraph or it contains an infinite independent set; in other words there exists an infinite subset of its vertices so that either every two of them are joined by an edge or no two of them are joined by an edge. Thus if we have a graph whose vertices are the integers, and which has no infinite complete sub-graph, it certainly has an infinite independent set. The question can now be asked if there exists an independent set whose vertices n1 < n2 < … do not tend to infinity too fast.

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