Connected-Intersecting Families of Graphs
For a graph property and a common vertex set , a family of graphs on is -intersecting iff satisfies for all in the family. Addressing a question of Chung, Graham, Frankl, and Shearer, we explore—for various —the maximum cardinality among all -intersecting families of graphs. In the connected-intersecting case, we resolve the question completely by a short linear algebraic proof showing this maximum is attained by taking all graphs containing a fixed spanning tree (though we show other extremal constructions as well). We also present a new lower bound for containing unions of a fixed subgraph.
One of the earliest and most important results in extremal combinatorics is the Erdős–Ko–Rado theorem , which states the following:
Theorem 1 (Erdős–Ko–Rado ).
Suppose and is a collection of -element subsets of such that for all , . Then . Moreover, for equality is attained iff is a collection of all -element sets containing some particular element.
Since its publication in 1961, the Erdős–Ko–Rado theorem has been generalized in many ways (e.g., via the equally classic Hilton–Milner theorem ), and it has spawned a host of related questions (see  as an example and [10, 8] for two recent surveys). In a 1986 publication , Chung, Graham, Frankl, and Shearer posed a broad generalization of the problem requiring pairwise intersections to have certain structure. Of this, we recall only the case for graphs, which is our primary focus.
Suppose is a graph property and is a family of graphs each on some common vertex set . We say is -intersecting iff satisfies for all . In this language, the main question posed in  is the following:
Question A: For a fixed property , how large is the largest -intersecting family of graphs?
To establish notation, let
denote the maximum size of such a family. Note this count is normalized in our definition so that is the fraction of graphs on in . Thus, Question A is to bound .
Although Question A was posed in full generality, the majority of research to date has been for properties of the form “contains a copy of the graph .” Graph families for which each pairwise intersection contains some copy of are called -intersecting, and triangle-intersecting families were of particular interest. For any fixed (non-empty) graph , we have the almost immediate bounds
The upper bound holds since an intersecting family cannot contain both a graph and its complement, and the lower bound is attained by fixing a particular copy of and taking all graphs containing it.
In 1976, Simonovits and Sós conjectured that for triangle-intersecting families, the lower bound is actually correct. In fact, this conjecture was a motivation for , who proved (inter alia)
Their argument relies on entropy and is (essentially) the earliest use of what is now known as Shearer’s lemma. This upper bound remained unchanged for decades until a breakthrough of Ellis, Filmus, and Freidgut  using Fourier analytic techniques to show
This result completely settles the case for triangles, and  also shows that equality is attained only for the canonical (i.e., ‘naïve’) construction described above. They also obtain an analogous statement about triangle-intersecting families under more general product measures.
For other intersecting families, it is a folklore result—alluded to in  and fully stated by Alon and Spencer —that if is a constellation (i.e., a forest in which each connected component is a star) then . Alon and Spencer conjecture (as echoed in ) that for every other graph . This would follow from the case , the path with three edges, but in fact there are no known upper bounds for any bipartite that are asymptotically better than (1). And although it is natural to conjecture that in general the lower bound of (1) is tight, this was disproven by Christofides  who found a construction exhibitting
which is tight for even by taking all graphs containing some particular perfect matching. For any fixed , another modification of  also gives the upper bound , although we could not find any explicit reference for this.
Our main result is proven in Section 2, where we consider connected-intersecting families, settling the corresponding case of Question A completely.
Suppose is a connected-intersecting family of graphs on (i.e., each pairwise intersection is connected). Then . Therefore, .
Although this is tight via the canonical examples (i.e., all graphs containing some fixed spanning tree), we also exhibit many other graph families that are just as large. Our proof is a short linear algebraic argument, related to the idea of anti-clusters as introduced by Griggs and Walker .
For any non-empty graph , let be the graph consisting of vertex-disjoint copies of . For all , there exists an integer such that for each fixed
Note that for any strongly increasing property (i.e., any property retained both by the addition of edges as well as the addition of vertices) the function is non-decreasing in , so the limits in Theorem 3 exist for any fixed . We conclude in Section 4 with open questions and conjectures.
Acknowledgement: We would like to thank the support and funding of the 2018 Summer Undergraduate Math Research at Yale (SUMRY) program, where much of this project was completed.
2 Connected intersections
Suppose is a connected-intersecting family. Following a standard approach (see, e.g., ), we equip the space of graphs on with an operation where denotes the graph on whose edge set is . This gives the set of graphs on the structure of an -dimensional vector space over .
For , let denote the complete bipartite graph on with edges iff . Note that the set consisting of all complete bipartite graphs is a subspace: in fact . Moreover, since iff or .
Finally, we claim that intersects each coset of at most once. Otherwise, has distinct elements of the form and . Because , we know that is the complement of two non-empty cliques, and hence is disconnected. But for any graphs , which would imply is disconnected. Thus, since intersects each coset of at most once, . ∎
If is any spanning tree, then the proportion of graphs containing is precisely , which yields a connected-intersecting family of maximum possible size. However, there are many more families of this same size. For example, let and be vertex-disjoint trees whose union spans every vertex of and let be a subset of edges between and such that is odd. Take to be the family of graphs containing as well as at least half the edges of . Then is connected-intersecting since the intersection of any two elements contains as well as at least one edge of (since is odd), and is exactly of the graphs. This construction can also be iterated to obtain large, strange families of connected-intersecting graphs. Moreover, it is not difficult to find extremal connected-intersecting families not of this type on as few as four vertices (e.g., for , take a -cycle and also all graphs having at least 5 edges).
3 Containing disjoint unions
We now turn our attention to a proof of Theorem 3. In keeping with the notation of the theorem, let be a fixed non-empty graph and let denote the graph consisting of vertex-disjoint copies of . Our construction relies on the following simple “tensoring” trick.
Namely, suppose is a graph family on , and consider a graph on . For , let be the subgraph of induced on the vertex set . As a slight abuse of notation, we say iff the graph isomorphic to obtained by subtracting from the label of each vertex is an element of .
Finally, for integers , let denote the family of graphs on such that (i.e., graphs for which at least of the projections belong to ).
Suppose is an optimal -intersecting family of graphs on . Then for any , is a -intersecting family (since for any two elements of , there are at least indices for which both projections are elements of ). Therefore, we have
Setting and taking limits of the above as , we obtain
where denotes the binary logarithm. For each value of , we then pick so that is as close as possible to . The fact that as then ensures that the terms within the braces above tend to by continuity.
The most obvious open problem is to resolve Question A for other , of which increasing properties are especially natural (e.g., containment of fixed subgraphs, high chromatic number, non-planarity). The most famous such problem is to find a bound , but a result of the form for any particular bipartite would likely be great progress.
Closer to the results of this paper, we suggest Question A in the case of Hamiltonicity. By Theorem 2 (and from the canonical lower bound of fixing a particular Hamiltonian cycle), we have
Here, we conjecture that the lower bound is tight, but it is unclear how to improve either bound.
It would also be interesting to study , the path with edges. For , Theorem 2 shows this is , and we have already discussed the case . A natural next step might be to study for other fixed and ask when . We also suggest the properties “has at most connected components” and also “has minimum degree at least .”
Another direction of research could be to ask for quantitative improvements in Theorem 3. That is, for a fixed graph , let . A tensoring construction similar to that of Section 3 shows , so by Fekete’s lemma, exists. Theorem 3 proves this limit is at least , and it is natural to ask whether or not this is optimal. However, considering the weighted version of Katona’s -intersection theorem (see  for a solution and discussion, and see  for an elegant related coupling argument), we believe that any improvement to our lower bound would require a rather different approach.
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Massachusetts Institute of Technology