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### Some intersection theorems for ordered sets and graphs

F.R.K Chung, R.L Graham, P Frankl, J.B Shearer
1986 Journal of combinatorial theory. Series A
We will now use the Product Theorem to prove two theorems on intersection families of graphs. THEOREM 8.  ...  Suppose [n] = S, u ... u Sk is a partition of [n] into k non-empty sets, and B G 2["' is a family with the property that for some j, 16 j f k, each 3 E B intersects at least j of the Si, 1 ,< i < k.  ...

### The Intersection Graph of Subgroups of the Dihedral Group of Order 2pq

Peshawa M. Khudhur, Rashad R. Haji, Sanhan M.S. Khasraw
2021 Iraqi Journal of Science
For a finite group G, the intersection graph of G is the graph whose vertex set is the set of all proper non-trivial subgroups of G, where two distinct vertices are adjacent if their intersection is a  ...  In this article, we investigate the detour index, eccentric connectivity, and total eccentricity polynomials of the intersection graph of subgroups of the dihedral group for distinct primes .  ...  Through this article, we fixed the sets , , and In this section, some basic properties of the intersection graph of are investigated, such Thus, ( ) , for all and { }. . ( ) as the order and chromatic  ...

### Polynomial treewidth forces a large grid-like-minor

Bruce A. Reed, David R. Wood
2012 European journal of combinatorics (Print)
A grid-like-minor of order ℓ in a graph G is a set of paths in G whose intersection graph is bipartite and contains a K_ℓ-minor.  ...  For example, the rows and columns of the ℓ×ℓ grid are a grid-like-minor of order ℓ+1. We prove that polynomial treewidth forces a large grid-like-minor.  ...  A grid-like-minor of order ℓ in a graph G is a set P of paths in G, such that the intersection graph 2 of P is bipartite and contains a K ℓ -minor.  ...

### Algorithms for some intersection graphs [chapter]

T. Kashiwabara
1981 Lecture Notes in Computer Science
Several intersection graphs such as curves-in-the-plane graphs, circular-arc graphs, chordal graphs and interval graphs are reviewed, especially on their recognition algorithms.  ...  In this connection graph realization problem is mentioned.  ...  Theorem 7 v n, Vn_l,...,v I is a chordal order of vertices iff X(Vn)> X(Vn_l)>...>x(v I) for some rooted tree model.  ...

### Regular bipartite graphs and intersecting families [article]

Andrey Kupavskii, Dmitriy Zakharov
2017 arXiv   pre-print
In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H os--Ko--Rado theorem, the Hilton--Milner theorem,  ...  a theorem due to Frankl concerning the size of intersecting families with bounded maximal degree, and versions of results on the sum of sizes of non-empty cross-intersecting families due to Frankl and  ...  We would like to thank Peter Frankl and the anonymous referee for bringing several references to our attention, and for useful comments on the manuscript that helped to improve the presentation.  ...

### Largest size and union of Helly families

Zsolt Tuza
1994 Discrete Mathematics
for large n, ~~(<(";~;')+(~~~)+ 1, and the set system attaining equality is unique.  ...  The main tool is a result (an analogue of the Hilton-Mimer theorem on intersecting k-uniform set systems) stating that if the sets in a k-uniform Helly family on n points have an empty intersection, then  ...  of order at least d"n (for some constant d") in the graph with edge set {S\W'ISEY, K'ISS W").  ...

### Steiner intervals in graphs

Ewa Kubicka, Grzegorz Kubicki, Ortrud R. Oellermann
1998 Discrete Applied Mathematics
Let G be a graph and U, L' two vertices of G. Then the interval from K to 2' consists of all those vertices that lie on some shortest u -1; path. Let S be a set of vertices in a connected graph G.  ...  Moreover. for every II > 4, those graphs with the property that the 3-intersection interval of every n-set is nonempty are completely characterized.  ...  Acknowledgements The authors are grateful to the referees for their detailed comments which greatly improved this article.  ...

### Some lower bounds for the L-intersection number of graphs [article]

Zeinab Maleki, Behnaz Omoomi
2013 arXiv   pre-print
In this paper, some lower bounds for the (bipartite) L-intersection number of a graph for various types L in terms of the minimum rank of graph are obtained.  ...  For a set of non-negative integers L, the L-intersection number of a graph is the smallest number l for which there is an assignment on the vertices to subsets A_v ⊆{1,..., l}, such that every two vertices  ...  From the above corollary, we could get lower bounds of order n for some graphs.  ...

### The intersection power graph associated with a finite group

Wei Lv, Xuanlong Ma
2021 ScienceAsia
Furthermore, we characterize the finite groups whose intersection power graphs equal to their power graphs, enhanced power graphs, commuting graphs, and order supergraphs.  ...  Moreover, the finite groups with dominatable intersection power graphs are characterized, which generalizes some results in Bera [Electron J Graph Theory Appl 6 (2018):178-189].  ...  As a result, the vertex set of the commuting graph of G is assumed to the set of all non-central elements of G. For some results on commuting graphs, see [9, 10] and references therein.  ...

### On Intersection Graph of Dihedral Group [article]

Sanhan Khasraw
2020 arXiv   pre-print
The intersection graph of G is a graph whose vertex set is the set of all proper non-trivial subgroups of G and two distinct vertices H and K are adjacent if and only if H∩ K ≠{e}, where e is the identity  ...  graph of D_2n for n=p^2, p is prime.  ...  Then 4 Metric dimension and resolving polynomial of intersection graph on D 2p 2 For a vertex u of a graph Γ, the set N(u) = {v ∈ V (Γ) : uv ∈ E(Γ)} is called the open neighborhood of u and the set N  ...

### Comparability graphs and intersection graphs

Martin Charles Golumbic, Doron Rotem, Jorge Urrutia
1983 Discrete Mathematics
It is also shown that G is the intersection graph of the concatenation of &c permutation diagrams if and only if the partial order dimension of e is ak t 1.  ...  We call the intersection graph of D a function graph (f-graph). It is shown that a graph G is an f-graph if and only if its complement 0 is a comparability graph.  ...  In addition, we study some connections between function graphs, permutation graphs, and the dimension of partially ordered sets.  ...

### Subgroup intersection graph of finite abelian groups

T. Tamizh Chelvam, M. Sattanathan
2012 Transactions on Combinatorics
The subgroup intersection graph Gamma_SI (G) of G isa graph with vertex set G − e and two distinct vertices x and y are adjacent if and only if | i ∩ | | > 1.  ...  In this paper, we obtain a lower bound for the independence number of subgroup intersection graph.  ...  The power graph [4, 2] of G is the graph with vertex set G and two vertices x and y are adjacent if either x = y i or y = x j for some positive integers i and j. In [8] , T. Tamizh Chelvam and M.  ...

### On cross-intersecting families of independent sets in graphs [article]

Vikram Kamat
2010 arXiv   pre-print
In particular we build on a result of Borg and Leader for signed sets and prove a theorem for uniform cross-intersecting subfamilies of independent vertex subsets of a disjoint union of complete graphs  ...  We formulate a graph-theoretic analogue of Hilton's cross-intersection theorem, similar to the one developed by Holroyd, Spencer and Talbot for the Erdos-Ko-Rado theorem.  ...  Let this ordering be [v 1 , . . . , v m ] where m = |V (H)| and let v 1 v i ∈ E(H) for some 2 ≤ i ≤ m. Let A and B be a cross-intersecting pair in J r (G).  ...

### Algorithmic aspects of intersection graphs and representation hypergraphs

Martin Charles Golumbic
1988 Graphs and Combinatorics
Let ~ be a family of sets. The intersection graph of ~ is obtained by representing each set in N' by a vertex and connecting two vertices by an edge if and only if their corresponding sets intersect.  ...  Of primary interest are those classes of intersection graphs of families of sets having some specific topological or other structure.  ...  A set A c X is called an articulation set for H if A = E1 f3 E 2 for some pair of hyperedges El, E2 ~ ~ and H[X -A] has more connected components than H.  ...

### Turán-type results for partial orders and intersection graphs of convex sets

Jacob Fox, János Pach, Csaba D. Tóth
2010 Israel Journal of Mathematics
Our bounds rely on new Turán-type results on incomparability graphs of partially ordered sets.  ...  There is a constant c > 0 such that for every family F of n convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn.  ...  It is very easy to see that it is sufficient to establish Theorems 1 and 2 for collections of sets intersecting the same line.  ...
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