Planar Graphs have Independence Ratio at least 3/13

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$. In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

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

Advances in Combinatorics ◽

10.19086/aic.13667 ◽

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.

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Non-Separating Planar Graphs

The Electronic Journal of Combinatorics ◽

10.37236/8816 ◽

2021 ◽

Vol 28 (1) ◽

Author(s):

Hooman R. Dehkordi ◽

Graham Farr

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Disjoint Union ◽

Disjoint Paths ◽

Outerplanar Graph ◽

Outerplanar Graphs ◽

Structural Characterisation ◽

A Minor ◽

Vertex Disjoint

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$. Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs. In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$ as a minor. Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a wheel or it is a graph obtained from the disjoint union of two triangles by adding three vertex-disjoint paths between the two triangles. Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges. Thus, maximal linkless graphs can have significantly fewer edges than maximum linkless graphs; Sachs exhibited linkless graphs with $n$ vertices and $4n-10$ edges (the maximum possible) in 1983.

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Path Choosability of Planar Graphs

The Electronic Journal of Combinatorics ◽

10.37236/7139 ◽

2018 ◽

Vol 25 (4) ◽

Author(s):

Glenn G. Chappell ◽

Chris Hartman

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Disjoint Union ◽

Vertex Coloring ◽

List Coloring ◽

Outerplanar Graphs ◽

Color Class ◽

Path Coloring

A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.

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A 9 k kernel for nonseparating independent set in planar graphs

Theoretical Computer Science ◽

10.1016/j.tcs.2013.11.021 ◽

2014 ◽

Vol 516 ◽

pp. 86-95 ◽

Cited By ~ 2

Author(s):

Łukasz Kowalik ◽

Marcin Mucha

Keyword(s):

Planar Graphs ◽

Independent Set

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Roman domination and 2-independence in trees

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830917500239 ◽

2017 ◽

Vol 09 (02) ◽

pp. 1750023 ◽

Cited By ~ 1

Author(s):

Nacéra Meddah ◽

Mustapha Chellali

Keyword(s):

Independence Number ◽

Minimum Weight ◽

Domination Number ◽

Independent Set ◽

Maximum Cardinality ◽

Roman Domination ◽

Roman Domination Number ◽

Number Formula ◽

Sum Formula ◽

Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

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The Weakly Nilpotent Graph of a Commutative Ring

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-096-1 ◽

2017 ◽

Vol 60 (2) ◽

pp. 319-328

Author(s):

Soheila Khojasteh ◽

Mohammad Javad Nikmehr

Keyword(s):

Commutative Ring ◽

Chromatic Number ◽

Disjoint Union ◽

Independence Number ◽

Artinian Ring ◽

Clique Number ◽

Commutative Rings ◽

Unicyclic Graph ◽

Nilpotent Elements ◽

Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

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KNOT-INEVITABLE PROJECTIONS OF PLANAR GRAPHS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216596000485 ◽

1996 ◽

Vol 05 (06) ◽

pp. 877-883 ◽

Cited By ~ 2

Author(s):

KOUKI TANIYAMA ◽

TATSUYA TSUKAMOTO

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Spatial Graph ◽

Torus Knot ◽

Nonplanar Graph

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2, n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.

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r-Hued coloring of planar graphs without short cycles

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500342 ◽

2020 ◽

Vol 12 (03) ◽

pp. 2050034

Author(s):

Yuehua Bu ◽

Xiaofang Wang

Keyword(s):

Planar Graph ◽

Chromatic Number ◽

Planar Graphs ◽

Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

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