Graph Operations and Upper Bounds on Graph Homomorphism Counts Luke Sernau March 9, 2017 Abstract We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any n-vertex, d-regular graph G and any graph H (possibly with loops), n n d d hom(G, H) ≤ max hom(Kd,d,H) 2 , hom(Kd+1,H) +1 , n o where hom(G, H) is the number of homomorphisms from G to H. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of H for which the bound stated above on hom(G, H) holds for all n-vertex, d-regular G. In particular we show that if HWR is the complete looped path on three vertices, also known as the Widom-Rowlinson graph, then n d hom(G, HWR) ≤ hom(Kd+1,HWR) +1 for all n-vertex, d-regular G. This verifies a conjecture of Galvin. arXiv:1510.01833v3 [math.CO] 8 Mar 2017 1 Introduction Graph homomorphisms are an important concept in many areas of graph theory. A graph homomorphism is simply an adjacency-preserving map be- tween a graph G and a graph H. That is, for given graphs G and H, a function φ : V (G) → V (H) is said to be a homomorphism from G to H if for every edge uv ∈ E(G), we have φ(u)φ(v) ∈ E(H) (Here, as throughout, all graphs are simple, meaning without multi-edges, but they are permitted 1 to have loops). When H = Kq, the complete loop-free graph on q vertices, a homomorpism from G to H is just a proper q-coloring of G. For this reason homomorphisms from G to H are often referred to as H-colorings of G. While graph homomorphisms have proven to be a very useful tool, many extremal problems involving them remain open [2]. One such question is the following. Question 1. Let H be an arbitrary graph. Fix natural numbers d and n, with d < n. What d-regular graphs G on n vertices maximize hom(G,H), the number of homomorphisms from G to H? This question remains open, but one conjecture (first given in [5]) is that the maximizer graph for any H is one of two possibilities. Conjecture 1. Fix an arbitrary graph H. Let G be a loop-free d-regular graph on n vertices. Then n n d d hom(G,H) ≤ max hom(Kd,d,H) 2 , hom(Kd+1,H) +1 . n o Note that if 2d divides n, n n d hom(Kd,d,H) 2 = hom Kd,d,H , 2d n n where by 2d Kd,d we mean the disjoint union of 2d copies of Kd,d. Likewise, if d + 1 divides n, n n hom(K ,H) d+1 = hom K ,H . d+1 d +1 d+1 So Conjecture 1 is really making a claim about maximizer graphs, rather than merely asserting an upper bound. Namely, it asserts that up to some divisibility considerations, the maximizer of the number of homomorphisms is either the disjoint union of an appropriate number of copies of Kd,d or the disjoint union of an appropriate number of copies of Kd+1. This conjecture arose as a correction to an earlier conjecture (made in [6]), that hom(G,H) ≤ n d hom(Kd,d,H) 2 , always. In [6], Galvin and Tetali were able to show this stronger form for the special case where G is bipartite, and so it was natural to consider generalizing the result to all graphs. But taking H to be the disjoint union of loops gives a case where disjoint copies of Kd+1 is the maximizer, and so we are forced to consider the weaker form given in Conjecture 1. 2 The first aim of this article is to show that Conjecture 1 is false. Indeed, in Section 3 we give counterexamples for all d ≥ 4 (see Theorem 5 for the exact statement). It is now not clear what the correct version of Conjecture 1 should be for general H. In the absence of a general statement, one sensible avenue of n n approach is to consider conditions on H under which one of 2d Kd,d, d+1 Kd+1 is the maximizer of the number of homomorphisms for all G. In [10], Zhao n/2d gave an infinite family of H for which hom(G,H) ≤ hom(Kd,d,H) for all n-vertex, d-regular G, and in [5] Galvin exhibited infinitely many triples n/(d+1) (n,d,H) for which hom(G,H) ≤ hom(Kd+1,H) for all n-vertex, d- regular G. The second aim of this article is to continue this line of inves- tigation. We develop the properties of two graph operations, and then use them to establish a large family of choices of H for which some number of copies of Kd,d is the maximizer, as well as another large family for which an appropriate number of copies of Kd+1 is the maximizer. Specifically, we prove the following. Theorem 1. Let H1 and H2 be graphs with the property that for all n and d n/2d and all n-vertex, d-regular G, we have hom(G,Hi) ≤ hom(Kd,d,Hi) . Let A be an arbitrary graph, and B an arbitrary bipartite graph. If H˜ is any of A B H1 × H2, H1 , or A , then for all n and d and all n-vertex, d-regular G, we have n/2d hom(G, H˜ ) ≤ hom(Kd,d, H˜ ) . (1) B The operations of tensor product H1 × H2 and graph exponentiation A will be defined formally in Section 4. As an example of Theorem 1, consider the graph H consisting of one edge, with one endvertex looped. For this choice of H, it is easy to see that hom(G,H) is the number of independent sets in G (that is, the number of sets of mutually non-adjacent vertices). Zhao [11] showed that in this case n d hom(G,H) ≤ hom(Kd,d,H) 2 holds for all G. Now applying Theorem 1 we find that (1) holds for all n-vertex d-regular G with H˜ = H × H. This graph turns out the be the graph on four vertices composed of a triangle with one vertex looped, and an additional vertex adjacent to the looped vertex. It’s worth noting that this is not a graph that can be addressed using the methods Zhao develops in [10]. Zhao provides the following sufficient condition. 3 Theorem 2. For an arbitrary graph H, construct a graph Hbst with vertices V (Hbst)= V (H)×V (H) and an edge between (u, v) and (u′, v′) ∈ V (Hbst) if and only if uu′ ∈ E(H), vv′ ∈ E(H), and either uv′ ∈/ E(H) or u′v∈ / E(H). If Hbst is bipartite, then for all n and d and all n-vertex, d-regular G, n/2d hom(G,H) ≤ hom(Kd,d,H) . (2) Our choice of H˜ does not meet this criterion, because H˜ bst contains an odd cycle (the construction is straightforward, and has been omitted). We also make use of the tensor product and the graph exponent to es- tablish a family of choices of H for which hom(G,H) is upper bounded by n d hom(Kd+1,H) +1 . Theorem 3. Let G be an arbitrary d-regular graph on n vertices. For an arbitrary graph H and bipartite graph B, n B B d hom(G,l(H )) ≤ hom(Kd+1,l(H )) +1 where l(G) denotes the induced subgraph of G formed by the set of looped vertices of G. When H is a path on two vertices with a loop on the first, and B is a single edge, l(HB) is easily seen to be a path on three vertices with a loop at each vertex. This graph, which we will call HWR, is often referred to as the Widom-Rowlinson graph and it encodes a well-studied model in statistical physics (see [1] for a good discussion). Galvin conjectured (implicitly in [5] n/(d+1) and explicitly in [4]) that hom(G,HWR) ≤ hom(Kd+1,HWR) for all n-vertex d-regular G, making it the simplest non-trivial example of an H for which unions of Kd+1 maximize the homomorphism count among regular graphs, and Theorem 3 verifies this conjecture. In an earlier version of this paper (arXiv:1510.01833v1) the right-hand B n/d side of the inequality in Theorem 3 was the weaker hom(Kd,l(H )) . This turned out to be the result of a small error in the proof. We are grateful to E. Cohen (personal communication) for pointing out this error. After the earlier version of this paper appeared, a separate result that also implies Galvin’s conjecture was obtained by Cohen, Perkins and Tetali [3]. In Section 2 we recall some results that will be important in what follows. In Section 3 we give a construction providing counterexamples to Conjecture 1. The results from Section 2 are then built upon in Section 4, culminating in the proof of Theorem 1. Lastly, in Section 5, we continue the discussion of exponential graphs in order to prove Theorem 3. 4 2 Foundational Results Before delving into the main body of work, it will be useful to recall a hand- ful of important results that we will draw on heavily.