‎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 of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

‎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 of subsets $A_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...

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 of subsets Av ⊆ {1, . . . , l} to vertices v, such that every two vertices u, v are adjacent if and only if |Au ∩ Av| ∈ L. The bipartite L-intersection number is defined similarly when the conditions are considered only for the vertices in different parts. In this...

let $g$ be a finite group with the identity $e$‎. ‎the subgroup intersection graph $gamma_{si}(g)$ of $g$ is the graph with vertex set $v(gamma_{si}(g)) = g-e$ and two distinct vertices $x$ and $y$ are adjacent in $gamma_{si}(g)$ if and only if $|leftlangle xrightrangle capleftlangle yrightrangle|>1$‎, ‎where $leftlangle xrightrangle $ is the cyclic subgroup of $g$ generated by $xin g$‎. ‎in th...

The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<infty)$, then the number of reducible $M$-ideals does not exceed of $frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The...

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line L if the intersection of its any member with L is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.

The sphericity sph(G) of a graph G is the minimum dimension d for which G is the intersection graph of a family of congruent spheres in Rd . The edge clique cover number (G) is the minimum cardinality of a set of cliques (complete subgraphs) that covers all edges of G. We prove that if G has at least one edge, then sph(G) (G). Our upper bound remains valid for intersection graphs defined by bal...

In this paper we study restrictions imposed on individual preferences, as opposed to preference profiles, and we introduce individual preference restrictions that characterize the existence and uniqueness of stable hedonic coalition structures on rich preference domains. We prove that if such a preference domain satisfies an individual preference restriction then it guarantees the existence of ...