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Some intersection theorems for ordered sets and graphs

1986
*
Journal of combinatorial theory. Series A
*

We will now use the Product

doi:10.1016/0097-3165(86)90019-1
fatcat:jyv3oagdk5fd5azhemeu5janru
*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. ...##
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The Intersection Graph of Subgroups of the Dihedral Group of Order 2pq

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 ...

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

2012
*
European journal of combinatorics (Print)
*

A grid-like-minor of

doi:10.1016/j.ejc.2011.09.004
fatcat:3jcng4dsxrce3hmhpuzo2fmvla
*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. ...##
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Algorithms for some intersection graphs
[chapter]

1981
*
Lecture Notes in Computer Science
*

Several

doi:10.1007/3-540-10704-5_15
fatcat:xzfteaeh5bavpj2bibjkserhzq
*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]

2017
*
arXiv
*
pre-print

In this paper we present a simple unifying approach to prove several statements about

arXiv:1611.03129v2
fatcat:kuq4weciovatrhlev2lyjaznma
*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. ...##
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Largest size and union of Helly families

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

1998
*
Discrete Applied Mathematics
*

Let G be a

doi:10.1016/s0166-218x(97)00084-x
fatcat:vtw33pp7fzegpiihe4cmlx6zwy
*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]

2013
*
arXiv
*
pre-print

In this paper,

arXiv:1211.0328v3
fatcat:mhiu6dpbxjhdvjv2ltmpezonzy
*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

2021
*
ScienceAsia
*

Furthermore, we characterize the finite groups whose

doi:10.2306/scienceasia1513-1874.2021.091
fatcat:ictcab3m7zfvtnv6l7vsb6zpta
*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. ...##
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On Intersection Graph of Dihedral Group
[article]

2020
*
arXiv
*
pre-print

The

arXiv:2011.10544v3
fatcat:go76evsz75bidmsrzkq3tahppy
*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 ...##
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Comparability graphs and intersection graphs

1983
*
Discrete Mathematics
*

It is also shown that G is the

doi:10.1016/0012-365x(83)90019-5
fatcat:7btcmvun35dcnhnvs6wykpblry
*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

2012
*
Transactions on Combinatorics
*

The subgroup

doaj:8d8bdb23b4074e0199ee55449e3a3621
fatcat:kutfn3mkxrgqtn7sm4xvzogthy
*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. ...##
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On cross-intersecting families of independent sets in graphs
[article]

2010
*
arXiv
*
pre-print

In particular we build on a result of Borg

arXiv:1010.0947v1
fatcat:jza5blmjd5gf5bjlhr7snoa6uy
*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

1988
*
Graphs and Combinatorics
*

Let ~ be a family of

doi:10.1007/bf01864170
fatcat:hc2zljrxq5hvdejv4d45vlg5wm
*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. ...##
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Turán-type results for partial orders and intersection graphs of convex sets

2010
*
Israel Journal of Mathematics
*

Our bounds rely on new Turán-type results on incomparability

doi:10.1007/s11856-010-0056-3
fatcat:njzhjej6eneczccxelopwifmmu
*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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