scholarly journals Spanning 3-Ended Trees in Almost Claw-Free Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Xiaodong Chen ◽  
Meijin Xu ◽  
Yanjun Liu
Keyword(s):  
Independent Set ◽  
Free Graph

We prove that ifGis ak-connected (k≥2) almost claw-free graph of ordernandσk+3(G)≥n+2k-2, thenGcontains a spanning 3-ended tree, whereσk(G)=min⁡{∑v∈S‍deg⁡(v):Sis an independent set ofGwithS=k}.

scholarly journals Graphs on Affine and Linear Spaces and Deuber Sets

10.37236/2732 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
David S. Gunderson ◽  
Hanno Lefmann
Keyword(s):  
Abelian Group ◽  
Finite Field ◽  
Recent Work ◽  
Independent Set ◽  
Large Set ◽  
Free Graph ◽  
Linear Spaces ◽  
Ramsey’S Theorem ◽  
Positive Integers ◽  
Arbitrary Abelian Group

If $G$ is a large $K_k$-free graph, by Ramsey's theorem, a large set of vertices is independent. For graphs whose vertices are positive integers, much recent work has been done to identify what arithmetic structure is possible in an independent set. This paper addresses  similar problems: for graphs whose vertices are affine or linear spaces over a finite field,  and when the vertices of the graph are elements of an arbitrary Abelian group.

scholarly journals Recognition and Optimization Algorithms for P5-Free Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 304
Author(s):  
Mihai Talmaciu ◽  
Luminiţa Dumitriu ◽  
Ioan Şuşnea ◽  
Victor Lepin ◽  
László Barna Iantovics
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Independent Set ◽  
Clique Number ◽  
Recognition Algorithm ◽  
Feedback Vertex Set ◽  
Free Graph ◽  
Dominating Number ◽  
Vertex Set ◽  
Np Complete

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

scholarly journals THE TYPICAL STRUCTURE OF MAXIMAL TRIANGLE-FREE GRAPHS

2015 ◽  
Vol 3 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
HONG LIU ◽  
ŠÁRKA PETŘÍČKOVÁ ◽  
MARYAM SHARIFZADEH
Keyword(s):  
Perfect Matching ◽  
Independent Set ◽  
Independent Sets ◽  
Free Graph ◽  
Typical Structure ◽  
Vertex Partition ◽  
Maximal Independent Sets ◽  
Vertex Disjoint ◽  
Removal Lemma

Recently, settling a question of Erdős, Balogh, and Petříčková showed that there are at most $2^{n^{2}/8+o(n^{2})}$$n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here, we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set.Our proof uses the Ruzsa–Szemerédi removal lemma, the Erdős–Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_{3}$s, which is of independent interest.

scholarly journals Counting Independent Sets in Hypergraphs

2014 ◽  
Vol 23 (4) ◽  
pp. 539-550 ◽  
Author(s):  
JEFF COOPER ◽  
KUNAL DUTTA ◽  
DHRUV MUBAYI
Keyword(s):  
Recent Result ◽  
Lower Bounds ◽  
Independent Set ◽  
Independent Sets ◽  
Average Degree ◽  
General Statement ◽  
Free Graph ◽  
Hereditary Properties ◽  
Image Position ◽  
Linear Hypergraph

Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least ${\exp\biggl({1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t} \biggl(\frac{1}{2}\ln t-1\biggr)\biggr)}$ independent sets. This improves a recent result of the first and third authors [8]. In particular, it implies that as n → ∞, every triangle-free graph on n vertices has at least ${e^{(c_1-o(1)) \sqrt{n} \ln n}}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 = 0.208138 \ldots$. Further, we show that for all n, there exists a triangle-free graph with n vertices which has at most ${e^{(c_2+o(1))\sqrt{n}\ln n}}$ independent sets, where $c_2 = 2\sqrt{\ln 2} = 1.665109 \ldots$. This disproves a conjecture from [8].Let H be a (k+1)-uniform linear hypergraph with n vertices and average degree t. We also show that there exists a constant ck such that the number of independent sets in H is at least ${\exp\biggl({c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}\biggr})}.$ This is tight apart from the constant ck and generalizes a result of Duke, Lefmann and Rödl [9], which guarantees the existence of an independent set of size $\Omega\biggl(\frac{n}{t^{1/k}} \ln^{1/k}t\biggr).$ Both of our lower bounds follow from a more general statement, which applies to hereditary properties of hypergraphs.

scholarly journals Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

2021 ◽  
Author(s):  
Raffaele Mosca
Keyword(s):  
Polynomial Time ◽  
Disjoint Union ◽  
Independent Set ◽  
Maximal Independent Set ◽  
Induced Subgraphs ◽  
Free Graph ◽  
Graph Classes ◽  
Graph Property ◽  
Np Hard Problem ◽  
Induced Path

AbstractThe Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs G and H, $$G+H$$ G + H denotes the disjoint union of G and H. This manuscript shows that (i) WIS can be solved for ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graphs in polynomial time, where a $$P_4$$ P 4 is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graph G there is a family $${{\mathcal {S}}}$$ S of subsets of V(G) inducing (complete) bipartite subgraphs of G, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of G is contained in some member of $${\mathcal {S}}$$ S . These results seem to be harmonic with respect to other polynomial results for WIS on [subclasses of] certain $$S_{i,j,k}$$ S i , j , k -free graphs and to other structure results on [subclasses of] Triangle-free graphs.

scholarly journals Improved bounds for the Erdős-Rogers function

2020 ◽  
Author(s):  
Timothy Gowers ◽  
Oliver Janzer
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Related Problem ◽  
Independent Set ◽  
Large Structure ◽  
Induced Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Subgraph

[Ramsey's Theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large "tame" substructure. The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{\alpha_s}$ vertices with $\alpha_s<1/2$ for every $s\ge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2\le t\le 2s-1$ and $s\ge 3$.

scholarly journals Structure and Colour in Triangle-Free Graphs

10.37236/9267 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
N. R. Aravind ◽  
Stijn Cambie ◽  
Wouter Cames van Batenburg ◽  
Rémi De Joannis de Verclos ◽  
Ross J. Kang ◽  
...  
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Short Proof ◽  
Independent Set ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Constant Multiple ◽  
Free Graph ◽  
Related Notion ◽  
Induced Path

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $\chi$ contains an induced cycle of length $\Omega(\chi\log\chi)$ as $\chi\to\infty$. Even if one only demands an induced path of length $\Omega(\chi\log\chi)$, the conclusion would be sharp up to a constant multiple. We prove it for regular girth $5$ graphs and for girth $21$ graphs. As a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some $c >0$ such that for every forest $H$ on $D$ vertices, every triangle-free and induced $H$-free graph has chromatic number at most $c D/\log D$. We prove this assertion with 'triangle-free' replaced by 'regular girth 5'.

scholarly journals On-line coloring of $I_s$-free graphs

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Iwona Cieslik ◽  
Marcin Kozik ◽  
Piotr Micek
Keyword(s):  
Greedy Algorithm ◽  
Planar Graphs ◽  
Greedy Algorithms ◽  
Independent Set ◽  
Vertex Coloring ◽  
Tight Bound ◽  
Free Graph ◽  
Coloring Problem ◽  
International Audience ◽  
On Line

International audience An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of $I_s$-free graphs, i.e., graphs in which the maximal size of an independent set is at most $s-1$. An old Szemerédi's result implies that for each on-line algorithm A there exists an on-line presentation of an $I_s$-free graph $G$ forcing A to use at least $\frac{s}{2}χ ^{(G)}$ colors. We prove that any greedy algorithm uses at most $\frac{s}{2}χ^{(G)}$ colors for any on-line presentation of any $I_s$-free graph $G$. Since the class of co-planar graphs is a subclass of $I_5$-free graphs all greedy algorithms use at most $\frac{5}{2}χ (G)$ colors for co-planar $G$'s. We prove that, even in a smaller class, this is an almost tight bound.

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