On complete subgraphs of different orders

1976 ◽  
Vol 79 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Intersection Graph ◽  
Edge Disjoint ◽  
Vertex Set

Let S be a set and let {X1, …, Xn} = be a family of distinct subsets of S. The intersection graph Ω() of has vertex set {X1, …, Xn} and XiXj (i ≠ j) is an edge of Ω() if and only if Xi ∩ Xi ≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph has n ≥ 4 vertices then one can find a representing set with at most [n2/4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph with n ≥ 1 vertices can be covered with at most [n2/4] edge disjoint triangles and edges. We say that a graph G is covered with the subgraphs G1, …, Gk, if every edge of G is in at least one Gi. One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4).

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
2021 ◽  
Author(s):  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Polynomial Time ◽  
Disjoint Paths ◽  
Structural Parameter ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Edge Disjoint ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
Author(s):  
T. ASIR ◽  
T. TAMIZH CHELVAM
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Intersection Graph ◽  
Vertex Connectivity ◽  
Total Graph ◽  
Graph Theoretic ◽  
Vertex Set ◽  
Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

The Generalized Connectivity of Generalized Petersen Graph

2021 ◽  
Author(s):  
Yuan Si ◽  
Ping Li ◽  
Yuzhi Xiao ◽  
Jinxia Liang
Keyword(s):  
Steiner Trees ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Edge Disjoint ◽  
Generalized Connectivity ◽  
Vertex Set

For a vertex set [Formula: see text] of [Formula: see text], we use [Formula: see text] to denote the maximum number of edge-disjoint Steiner trees of [Formula: see text] such that any two of such trees intersect in [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. We get that for any generalized Petersen graph [Formula: see text] with [Formula: see text], [Formula: see text] when [Formula: see text]. We give the values of [Formula: see text] for Petersen graph [Formula: see text], where [Formula: see text], and the values of [Formula: see text] for generalized Petersen graph [Formula: see text], where [Formula: see text] and [Formula: see text].

scholarly journals A Note on the Linear Cycle Cover Conjecture of Gyárfás and Sárközy

10.37236/7329 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Beka Ergemlidze ◽  
Ervin Győri ◽  
Abhishek Methuku
Keyword(s):  
Independent Set ◽  
Uniform Hypergraph ◽  
Edge Disjoint ◽  
Cycle Cover ◽  
Vertex Set ◽  
Cyclic Sequence

A linear cycle in a $3$-uniform hypergraph $H$ is a cyclic sequence of hyperedges such that any two consecutive hyperedges intersect in exactly one element and non-consecutive hyperedges are disjoint. Let $\alpha(H)$ denote the size of a largest independent set of $H$.We show that the vertex set of every $3$-uniform hypergraph $H$ can be covered by at most $\alpha(H)$ edge-disjoint linear cycles (where we accept a vertex and a hyperedge as a linear cycle), proving a weaker version of a conjecture of Gyárfás and Sárközy.

The intersection graph of a group

2015 ◽  
Vol 14 (05) ◽  
pp. 1550065 ◽  
Author(s):  
S. Akbari ◽  
F. Heydari ◽  
M. Maghasedi
Keyword(s):  
Disjoint Union ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Free Graph ◽  
Intersection Graphs ◽  
Cyclic Groups ◽  
Vertex Set

Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ 1. In this paper, we show that the girth of Γ(G) is contained in the set {3, ∞}. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ(G) is triangle-free, then G is solvable. Also, we prove that if Γ(G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups.

On the intersection graph of gamma sets in the zero-divisor graph

2015 ◽  
Vol 07 (01) ◽  
pp. 1450067 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Selvakumar
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Intersection Graph ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring. The intersection graph of gamma sets in the zero-divisor graph Γ(R) of R is the graph IΓ(R) with vertex set as the collection of all gamma sets of the zero-divisor graph Γ(R) of R and two distinct vertices A and B are adjacent if and only if A ∩ B ≠ ∅. In this paper, we study about various properties of IΓ(R) and investigate the interplay between the graph theoretic properties of IΓ(R) and the ring theoretic properties of R.

scholarly journals REPARTITORS, SELECTORS AND SUPERSELECTORS

2006 ◽  
Vol 07 (03) ◽  
pp. 391-415 ◽  
Author(s):  
FRÉDÉRIC HAVET
Keyword(s):  
Lower Bounds ◽  
Disjoint Paths ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Vertex Set ◽  
Optimal Networks

An (n, p, f)-network G is a graph (V, E) where the vertex set V is partitioned into four subsets [Formula: see text] and [Formula: see text] called respectively the priorities, the ordinary inputs, the outputs and the switches, satisfying the following constraints: there are p priorities, n - p ordinary inputs and n + f outputs; each priority, each ordinary input and each output is connected to exactly one switch; switches have degree at most 4. An (n, p, f)-network is an (n, p, f)-repartitor if for any disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text] and [Formula: see text], there exist in G, n edge-disjoint paths, p of them from [Formula: see text] to [Formula: see text] and the n - p others joining [Formula: see text] to [Formula: see text]. The problem is to determine the minimum number R(n, p, f) of switches of an (n, p, f)-repartitor and to construct a repartitor with the smallest number of switches. In this paper, we show how to build general repartitors from (n, 0, f)-repartitors also called (n, n + f)-selectors. We then consrtuct selectors using more powerful networks called superselectors. An (n, 0, 0)-network is an n-superselector if for any subsets [Formula: see text] and [Formula: see text] with [Formula: see text], there exist in G, [Formula: see text] edge-disjoint paths joining [Formula: see text] to [Formula: see text]. We show that the minimum number of switches of an n-superselector S+ (n) is at most 17n + O(log(n)). We then deduce that [Formula: see text] if [Formula: see text], R(n, p, f) ≤ 18n + 34f + O( log (n + f)), if [Formula: see text] and [Formula: see text] if [Formula: see text]. Finally, we give lower bounds for R(n, 0, f) and S+ (n) and show optimal networks for small value of n.

scholarly journals Representations of Edge Intersection Graphs of Paths in a Tree

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Martin Charles Golumbic ◽  
Marina Lipshteyn ◽  
Michal Stern
Keyword(s):  
Analogous Result ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Network Applications ◽  
Host Trees ◽  
Complete Problems ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete ◽  
Simple Paths

International audience Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.

Intersection graphs of general linear groups

2020 ◽  
pp. 2150039
Author(s):  
Mai Hoang Bien ◽  
Do Hoang Viet
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
General Linear Group ◽  
Undirected Graph ◽  
Intersection Graph ◽  
General Linear ◽  
Linear Groups ◽  
Intersection Graphs ◽  
General Linear Groups ◽  
Vertex Set

Let [Formula: see text] be a field and [Formula: see text] the general linear group of degree [Formula: see text] over [Formula: see text]. The intersection graph [Formula: see text] of [Formula: see text] is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of [Formula: see text]. Two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] is a finite field containing at least three elements, then the diameter [Formula: see text] is [Formula: see text] or [Formula: see text]. We also classify [Formula: see text] according to [Formula: see text]. In case [Formula: see text] is infinite, we prove that [Formula: see text] is one-ended of diameter 2 and its unique end is thick.

The M-intersection graph of ideals of a commutative ring

2018 ◽  
Vol 10 (03) ◽  
pp. 1850038 ◽  
Author(s):  
F. Heydari
Keyword(s):  
Commutative Ring ◽  
Clique Number ◽  
Intersection Graph ◽  
Sufficient Condition ◽  
Semilocal Ring ◽  
Multiplication Formula ◽  
Vertex Set

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module, and let [Formula: see text] be the set of all nontrivial ideals of [Formula: see text]. The [Formula: see text]-intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. For every multiplication [Formula: see text]-module [Formula: see text], the diameter and the girth of [Formula: see text] are determined. Among other results, we prove that if [Formula: see text] is a faithful [Formula: see text]-module and the clique number of [Formula: see text] is finite, then [Formula: see text] is a semilocal ring. We denote the [Formula: see text]-intersection graph of ideals of the ring [Formula: see text] by [Formula: see text], where [Formula: see text] are integers and [Formula: see text] is a [Formula: see text]-module. We determine the values of [Formula: see text] and [Formula: see text] for which [Formula: see text] is perfect. Furthermore, we derive a sufficient condition for [Formula: see text] to be weakly perfect.

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