Some results on multithreshold graphs

04/30/2019 ∙ by Gregory J. Puleo, et al. ∙ 0

Jamison and Sprague defined a graph G to be a k-threshold graph with thresholds θ_1 , ..., θ_k (strictly increasing) if one can assign real numbers (r_v)_v ∈ V(G), called ranks, such that for every pair of vertices v,w, we have vw ∈ E(G) if and only if the inequality θ_i ≤ r_v + r_w holds for an odd number of indices i. When k=1 or k=2, the precise choice of thresholds θ_1, ..., θ_k does not matter, as a suitable transformation of the ranks transforms a representation with one choice of thresholds into a representation with any other choice of thresholds. Jamison asked whether this remained true for k ≥ 3 or whether different thresholds define different classes of graphs for such k, offering 50 for a solution of the problem. Letting C_t for t > 1 denote the class of 3-threshold graphs with thresholds -1, 1, t, we prove that there are infinitely many distinct classes C_t, answering Jamison's question. We also consider some other problems on multithreshold graphs, some of which remain open.

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1. Introduction

Multithreshold graphs were introduced by Jamison and Sprague [3] as a generalization of the well-studied threshold graphs, first introduced by Chvátal and Hammer [1]. Given real numbers with , we say that a simple graph is a -threshold graph with thresholds if there exist real numbers , called ranks, such that for every pair of distinct vertices , we have if and only if the inequality holds for an odd number of indices . (Equivalently, adopting the convention that , we want if and only if for some .) In this case, we call a -representation of .

We will abbreviate this notation by saying that is -threshold to mean that is -threshold with thresholds . When , we obtain the classical threshold graphs.

In the case of the classical threshold graphs, it is clear that the exact choice of threshold does not matter: by appropriately rescaling the vertex ranks, any -threshold graph is seen to also be a -threshold graph. The same observation holds for : any ranks witnessing that is -threshold can be transformed, via an appropriate affine transformation, into ranks witnessing that is -threshold.

At the 2019 Spring Sectional AMS Meeting in Auburn, Jamison asked whether this phenomenon continues for higher , and specifically whether it still holds when

. Observing that an affine transformation of the vertex ranks still uses up two “degrees of freedom” and let us express any

-threshold graph as a -threshold graph for some , his question can be phrased as follows.

Question 1 (Jamison).

Do there exist real numbers such that the class of -threshold graphs and the class of -threshold graphs differ?

Jamison offered a $50 bounty for an answer to this question. In this paper, we answer the question in the affirmative: letting denote the class of -threshold graphs, we prove in Section 2 that there are infinitely many distinct classes .

We also study some other questions involving multithreshold graphs. Say that is a -threshold graph if there exist real numbers such that is a -threshold graph. Jamison and Sprague [3] proved that for every graph , there is some such that is a -threshold graph. Thus, we may define the threshold number of a graph to be the smallest nonnegative such that is a -threshold graph.

It is natural to compare the parameter to other graph parameters involving threshold graphs. Cozzens and Leibowitz [2] define the threshold dimension of a graph to be the smallest nonnegative integer such that can be expressed as the union of threshold graphs. Since the complement of a threshold graph is a threshold graph, we can also view as the smallest nonnegative such that can be expressed as the intersection of threshold graphs.

Doignon observed, in a personal communication with the authors of [4], that any -threshold graph is the intersection of two threshold graphs, hence whenever . This observation suggests a possible converse:

Question 2 (Jamison).

Replacing with , does imply any bound on ?

Question 3 (Jamison).

Is bounded by any function of or of ?

Question 2 has a brief answer. For any graph and positive integer , let be the disjoint union of copies of . The graph evidently has , since is a forbidden induced subgraph for a threshold graph; on the other hand, is a -threshold graph, as witnessed by giving the endpoints and of the th edge ranks and . Hence there are graphs with for which is arbitrarily large.

In Section 3, we partially answer by proving that there are graphs with for which is arbitrarily large. Finally, in Section 4, we discuss some remaining open problems about multithreshold graphs, along the lines of the questions considered in this paper.

2. Distinct families of -threshold graphs

To facilitate proofs about multithreshold graphs, we introduce some notational conventions. Given a multithreshold representation of a graph , the weight of an edge or non-edge is the sum of the ranks of and . When it is understood which multithreshold representation we are working with, we will omit the function and simply write to stand for the rank of the vertex . (Hence, the weight of an edge will simply be written as .)

For positive integers , let .

Lemma 1.

For any and any , the graph is a -threshold graph.

Proof.

Write with . Letting be the endpoints of the th edge for , observe that the following ranks yield a -threshold representation of :

  • for ,

  • for .

Evidently for every edge . On the other hand, any nonadjacent pair of vertices has a weight whose absolute value is at least , hence does not fall into the interval , and whose value is at most , hence does not fall into the interval . Hence, this is a -representation of . ∎

Computational experiments suggest that this bound is sharp: that is not -threshold for any . Lacking a formal proof of this sharpness, we prove a weaker statement.

Lemma 2.

For integer , if is a -threshold graph then .

Proof.

View the edges whose weight lies in as colored red and view the edges whose weight lies in as colored yellow. Since the yellow edges form a threshold graph and is a forbidden induced subgraph for threshold graphs, there is at most one yellow edge in . Let , , …, be the edges of .

By symmetry, we may assume that for each and that . This implies that has the largest rank of all vertices and, thus, if there is a yellow edge, then that edge is .

Let if is red, and otherwise let , so that all edges are red.

Claim 1: whenever . If not, then there exist with and . Hence

and

which contradicts the fact that the edge is absent.

It follows that the intervals are nested, with .

Claim 2: and whenever . Using the previous claim, we have

hence since otherwise the edge should be present. Similarly, since , we have

hence since otherwise the edge should be present.

Claim 3: for all . We prove this by induction on . When this is just the assumption that . Assuming it holds for , we prove that it holds for . Observe that

and by the previous claim we have and , so that

Claim 4: for all . This follows immediately from the inequalities

Having established these claims, we now complete the proof. If , then Claim 4 gives and , so to avoid the unwanted edge , it is necessary that , which requires .

If , then Claim 4 gives , and since with , we have . Hence, to avoid the unwanted edge it is necessary that , which implies . ∎

Corollary 3.

For each , the graph is -threshold but not -threshold for any .

Corollary 4.

The classes for are pairwise distinct.

Corollary 5.

For all , there exist -threshold graphs that are not -threshold graphs.

3. Threshold number versus threshold dimension

In this section, we partially answer Question 3 by proving that there exist graphs with for which is arbitrarily large. We will require the following result of Cozzens and Leibowitz [2] concerning the threshold dimension of complete multipartite graphs:

Theorem 6 (Cozzens–Leibowitz [2]).

For positive integers , the complete -partite graph has threshold dimension .

Let . Applying Theorem 6 with all shows that . Therefore, to show that can be arbitrarily large for graphs with , it suffices to prove the following theorem.

Theorem 7.

If is a -threshold graph, then . In particular, .

Note that the first part of the theorem is stronger than the second part, which is obtained using a crude lower bound on .

To prove this theorem, we will use a lemma stated in terms of edge colorings (not necessarily proper) induced by a threshold representation. Given a -representation of , we assign colors to the edges of by giving edge color if its weight lies in the interval . (By the definition of a -representation, for each edge there is exactly one such .)

Now, given an edge coloring, we can view each triangle as inducing a multiset of colors on its edges (for example, we consider “ red edges and yellow edge” and “ red edge and yellow edges” as different multisets, despite having the same underlying set).

Lemma 8.

In a -representation of , no two triangles have the same multiset of colors appearing on their edges.

Before proving the lemma, we show how the proof of the theorem follows immediately.

Proof of Theorem 7.

When , both parts of the theorem clearly hold, since is required. For , observe that has an induced and thus is not a threshold graph; thus, we may assume .

Assume has a -representation. By Lemma 8, no two triangles have the same multiset of colors on their edges. Hence, by the pigeonhole principle, the number of triangles is at most the number of size- multisets from , which by the standard stars-and-bars argument is . Since , we have . Rearranging gives the desired inequality. ∎

Proof of Lemma 8.

Suppose to the contrary that two triangles and have the same multiset of colors on their edges. Without loss of generality, we may assume that:

  • ,

  • , and

  • .

Choose indices so that has weight in , has weight in , and has weight in . (For convenience, we will adopt the convention that .) Observe that forces .

Say an edge with weight in is red, an edge with weight in is yellow, and an edge with weight in is pink. (It is possible that some of these thresholds coincide, in which case an edge may be, say, both red and yellow.)

Since and the -edges have the same multiset of colors as the -edges, the colors of the -edges must agree with the colors of the corresponding -edges:

  • and are red,

  • and are yellow,

  • and are pink.

Now we will derive our contradiction using the absence of the -edges.

Claim 1: . If instead , then we have

forcing a red -edge, a contradiction.

Claim 2: . If instead , then since , we have

forcing a pink -edge, a contradiction.

Now since , we have

forcing a yellow -edge, again a contradiction. This completes the proof. ∎

4. Remarks and Open Questions

After being informed of a preliminary version of the results in Section 2, Jamison (personal communication) suggested studying the class , where is the class of -threshold graphs.

Intuition suggests that perhaps is related somehow to the class of -threshold graphs. Since is a -threshold graph for all , Lemma 2 implies that not all -threshold graphs lie in the class . On the other hand, since all -threshold graphs satisfy , and since Theorem 6 implies that , we see that is not a -threshold graph; however, , as the ranking in Figure 1 can easily be verified to be a -representation for whenever is sufficiently small (in terms of ). Thus, but is not -threshold; the two classes are incomparable.

Figure 1. -threshold ranking of .
Open Question 1.

Is there a nice characterization of the class ?

While the results in Section 4 imply that there are at least countably many distinct classes , it is not clear whether there are countably many distinct classes or uncountably many distinct classes. Indeed, it seems plausible that whenever are distinct real numbers exceeding .

Open Question 2.

Are there uncountably many distinct classes ?

Open Question 3.

Are there distinct real numbers such that ?

Theorem 7 only partially answers Question 3, which seeks a bound of in terms of or . In particular, the following questions remain open:

Open Question 4.

Is bounded on the class of graphs with ?

Open Question 5.

Is bounded by any function of ?

5. Acknowledgments

I thank Robert E. Jamison both for posing the original Question 1 that motivated this paper as well as the thought-provoking followup questions 2 and 3 after receiving a preliminary version of these results.

References

  • [1] Václav Chvátal and Peter L. Hammer, Aggregation of inequalities in integer programming, (1977), 145–162. Ann. of Discrete Math., Vol. 1. MR 0479384
  • [2] Margaret B. Cozzens and Rochelle Leibowitz, Threshold dimension of graphs, SIAM J. Algebraic Discrete Methods 5 (1984), no. 4, 579–595. MR 763986
  • [3] Robert E. Jamison and Alan P. Sprague, Multithreshold graphs, Manuscript submitted for publication, 2018.
  • [4] by same author, Bi-threshold permutation graphs, Manuscript submitted for publication, 2019.