Extremal H-colorings of trees and 2-connected graphs
aa r X i v : . [ m a t h . C O ] O c t Extremal H -colorings of trees and 2-connected graphs John Engbers ∗ David Galvin † February 28, 2018
Abstract
For graphs G and H , an H -coloring of G is an adjacency preserving map fromthe vertices of G to the vertices of H . H -colorings generalize such notions as inde-pendent sets and proper colorings in graphs. There has been much recent research onthe extremal question of finding the graph(s) among a fixed family that maximize orminimize the number of H -colorings. In this paper, we prove several results in thisarea.First, we find a class of graphs H with the property that for each H ∈ H , the n -vertex tree that minimizes the number of H -colorings is the path P n . We then presenta new proof of a theorem of Sidorenko, valid for large n , that for every H the star K ,n − is the n -vertex tree that maximizes the number of H -colorings. Our proof usesa stability technique which we also use to show that for any non-regular H (and certainregular H ) the complete bipartite graph K ,n − maximizes the number of H -coloringsof n -vertex 2-connected graphs. Finally, we show that the cycle C n has the most proper q -colorings among all n -vertex 2-connected graphs. For a simple loopless graph G = ( V ( G ) , E ( G )) and a graph H = ( V ( H ) , E ( H )) (possiblywith loops, but without multi-edges), an H -coloring of G is an adjacency-preserving map f : V ( G ) → V ( H ) (that is, a map satisfying f ( x ) ∼ H f ( y ) whenever x ∼ G y ). Denote byhom( G, H ) the number of H -colorings of G .The notion of H -coloring has been the focus of extensive research in recent years. Lov´asz’smonograph [11] explores natural connections to graph limits, quasi-randomness and propertytesting. Many important graph notions can be encoded via homomorphisms — for example, proper q -coloring using H = K q (the complete graph on q vertices), and independent (or stable ) sets using H = H ind (an edge with one looped endvertex). The language of H -coloring is also ideally suited for the mathematical study of hard-constraint spin models ∗ [email protected]; Department of Mathematics, Statistics and Computer Science, MarquetteUniversity, Milwaukee, WI 53201. Research supported by the Simons Foundation and by a MarquetteUniversity Summer Faculty Fellowship † [email protected]; Department of Mathematics, University of Notre Dame, Notre Dame IN 46556. Re-search supported by NSA grant H98230-13-1-0248, and by the Simons Foundation. k mutually repulsive particles, whichis encoded as an H -coloring by using the graph H = H WR ( k ) which has loops on every vertexof K ,k (the star on k + 1 vertices). Note that the original Widom-Rowlinson model [19] has k = 2.Many authors have addressed the following extremal enumerative question for H -coloring:given a family G of graphs, and a graph H , which G ∈ G maximizes or minimizes hom( G, H )?This question can be traced back to Birkhoff’s attacks on the 4-color theorem, but recentattention on it owes more to Wilf and (independently) Linial’s mid-1980’s query as to which n -vertex, m -edge graph admits the most proper q -colorings (i.e. has the most K q -colorings).For a survey of the wide variety of results and conjectures on the extremal enumerative H -coloring question, see [2].A focus of the present paper is the extremal enumerative H -coloring question for thefamily T ( n ), the set of all trees on n vertices. This family has two natural candidates forextremality, namely the path P n and the star K ,n − , and indeed in [12] Prodinger andTichy showed that these two are extremal for the count of independent sets in trees: for all T ∈ T ( n ), hom( P n , H ind ) ≤ hom( T, H ind ) ≤ hom( K ,n − , H ind ) . (1)The Hoffman-London matrix inequality (see e.g. [7, 10, 17]) is equivalent to the statementthat hom( P n , H ) ≤ hom( K ,n − , H ) for all H and n ; a significant generalization of this dueto Sidorenko [16] (see also [3] for a short proof) shows that the right-hand inequality of (1)extends to arbitrary H . Theorem 1.1 (Sidorenko) . Fix H and n ≥ . Then for any T ∈ T ( n ) , hom( T, H ) ≤ hom( K ,n − , H ) . In other words, the star admits not just the most independent sets among n -vertex trees,but also the most H -colorings for arbitrary H . Two points are worth noting here. First, sincedeleting edges in a graph cannot decrease the number of H -colorings, Theorem 1.1 showsthat among all connected graphs on n vertices, K ,n − admits the most H -colorings. Second,if we extend T ( n ) instead to the family of graphs on n vertices with minimum degree atleast 1 and consider even n , then as shown by the first author [5] the number of H -coloringsis maximized either by the star or by the graph consisting of a union of disjoint edges.The left-hand side of (1) turns out not to generalize to arbitrary H : Csikv´ari and Lin[3], following earlier work of Leontovich [9], exhibit a (large) tree H and a tree E on sevenvertices such that hom( E , H ) < hom( P , H ), and raise the natural question of characterizingthose H for which hom( P n , H ) ≤ hom( T, H ) holds for all n and all T ∈ T ( n ).Our first result gives a partial answer to this question. Before stating it, we need toestablish a convention concerning degrees of vertices in graphs with loops. Convention:
For all graphs in this paper, the degree of a vertex v is the number of neighborsof v , i.e., d ( v ) = |{ w : v ∼ w }| . In particular, a loop on a vertex adds one to the degree. Welet ∆ denote the maximum degree of H .We also let G ◦ denote the graph obtained from G by adding loops to every vertex in G .2 heorem 1.2. Let n ≥ and let H be a regular graph. For an integer ℓ ≥ , let H ◦ ( ℓ ) bethe join of H and K ◦ ℓ . Then for any T ∈ T ( n ) , hom( P n , H ◦ ( ℓ )) ≤ hom( T, H ◦ ( ℓ )) . Equality occurs if and only if T = P n or H ◦ ( ℓ ) = K ◦ q for some q ≥ ℓ . Notice that the result also holds for H where each component is of the form H ◦ ( ℓ ).Theorem 1.2 generalizes the left-hand side of (1), as H ind is the join of K and K ◦ , and ourproof is a generalization of the inductive approach used in [12].Since the Widom-Rowlinson graph H WR ( k ) can be constructed from the disjoint unionof k looped vertices by the addition of a single looped dominating vertex, an immediatecorollary of Theorem 1.2 is that for all n ≥ P n , H WR ( k )) ≤ hom( T, H WR ( k )) . for all T ∈ T ( n ). We also note in passing that Theorem 1.2 may be interpreted in termsof partial H -colorings of G (that is, H -colorings of induced subgraphs of G that need notbe extendable to H -colorings of G ). Specifically, when ℓ = 1 the theorem says that if H isregular, then among all n -vertex trees none admits fewer partial H -colorings than P n .Our second result is a new proof of Sidorenko’s theorem, valid for sufficiently large n . Theorem 1.3.
There is a constant c H such that if n ≥ c H and T ∈ T ( n ) then hom( T, H ) ≤ hom( K ,n − , H ) with equality if and only if H is regular or T = K ,n − . While Theorem 1.3 is weaker than Theorem 1.1 in that it only holds for n ≥ c H , it isnoteworthy for two reasons. Firstly, our proof for non-regular H uses a stability technique— we show that if a tree is not structurally close to a star (specifically, if it has a long path),then it admits significantly fewer H -colorings than the star, and if it is structurally almosta star but has some blemishes, then again it admits fewer H -colorings. Secondly, the proofis less tree-dependent than Sidorenko’s, and so the ideas used may be applicable in othersettings. We illustrate this by considering the extremal enumerative question for H -coloringsof 2-connected graphs. Let C ( n ) denote the set of 2-connected graphs on n vertices, andlet K a,b be the complete bipartite graph with a and b vertices in the two color sets. Also,for graph H with maximum degree ∆, denote by s ( H ) the number of ordered pairs ( i, j ) ofvertices of H satisfying | N ( i ) ∩ N ( j ) | = ∆. The special case H = H ind of the following wasestablished by Hua and Zhang [8]. Theorem 1.4.
For non-regular connected H there is a constant c H such that if n ≥ c H and G ∈ C ( n ) then hom( G, H ) ≤ hom( K ,n − , H ) with equality if and only if G = K ,n − .For ∆ -regular H the same conclusion holds whenever s ( H ) ≥ +1 (when H is looplessand bipartite) or s ( H ) ≥ ∆ + 1 (otherwise). H we have s ( H ) ≥ | V ( H ) | (consider ordered pairs of the form ( i, i )), soan immediate corollary in this case is that if | V ( H ) | ≥ + 1 (when H is loopless andbipartite) or if | V ( H ) | ≥ ∆ + 1 (otherwise) then for all large n the unique 2-connectedgraph with the most H -colorings is K ,n − . Note that the bounds on s ( H ) in the ∆-regularcase are tight with respect to the characterization of uniqueness: when H = K ◦ ∆ we havehom( G, H ) = ∆ n for all n -vertex G , and here s ( H ) = ∆ , and when H = K ∆ , ∆ (a looplessbipartite graph) we have hom( G, H ) = 2∆ n · { G bipartite } for all n -vertex connected graphs G , and here s ( H ) = 2∆ .Since all G ∈ C ( n ) are connected, the restriction to connected H is natural. Unlike withTheorem 1.3, the restriction to non-regular H is somewhat significant here. In particular,there are examples of regular graphs H such that for all large n , there are graphs in C ( n )that admit more H -colorings than K ,n − (and so there is no direct analog of Theorem 1.1in the world of 2-connected graphs). One such example is H = K ; it is easily checked thathom( C n , K ) > hom( K ,n − , K ) for all n ≥ C n is the cycle on n vertices). Moregenerally, we have the following. Theorem 1.5.
Let n ≥ and q ≥ be given. Then for any G ∈ C ( n ) , hom( G, K q ) ≤ hom( C n , K q ) , with equality if and only if G = C n , unless q = 3 and n = 5 in which case G = K , alsoachieves equality.The same conclusion holds with C ( n ) replaced by the larger family of -edge-connectedgraphs on n vertices. In the setting of Theorem 1.5 it turns out that no extra complications are introduced inmoving from 2-connected to 2-edge-connected. We do not, however, currently see a way toextend Theorem 1.4 to this larger family. In particular, the key structural lemma given inCorollary 4.4 does not generalize to the family of 2-edge-connected graphs.The rest of the paper is laid out as follows. The proof of Theorem 1.2 is given in Section2. We prove Theorem 1.3 in Section 3, and use similar ideas to then prove Theorem 1.4 inSection 4. The proof of Theorem 1.5 is given in Section 5. Finally, we end with a number ofopen questions in Section 6.
Let H be a ∆-regular graph, and let H ◦ ( ℓ ) be the join of H and K ◦ ℓ . Using induction on n , we will show something a little stronger than Theorem 1.2, namely that for any n -vertex forest F , hom( F, H ◦ ( ℓ )) ≥ hom( P n , H ◦ ( ℓ )), with equality if and only if either H ◦ ( ℓ ) is acomplete looped graph, or F = P n . We will first prove the inequality, and then address thecases of equality.For n ≤
4, the only forest F on n vertices that is not a subgraph of P n (and so for whichhom( F, H ◦ ( ℓ )) ≥ hom( P n , H ◦ ( ℓ )) is immediate) is F = K ,n − ; but in this case the requiredinequality follows directly from the Hoffman-London inequality [7, 10].So now fix n ≥
5, and let F be a forest on n vertices. In what follows, for x ∈ V ( F ) and i ∈ V ( H ◦ ( ℓ )) we write hom( F, H ◦ ( ℓ ) | x i ) for the number of homomorphisms from F to4 ◦ ( ℓ ) that map x to i . Also, we’ll assume that H ◦ ( ℓ ) has q + ℓ vertices, say V ( H ◦ ( ℓ )) = { , . . . , q, . . . , q + ℓ } , with the looped dominating vertices being q + 1 , . . . , q + ℓ .Let x be a leaf in F , with unique neighbor y (note that we may assume that F has anedge, because the desired inequality is trivial otherwise). For each i ∈ { , . . . , q + ℓ } ,hom( F, H ◦ ( ℓ ) | x i ) = X j ∼ i hom( F − x, H ◦ ( ℓ ) | y j ) , (2)which in particular implies hom( F, H ◦ ( ℓ ) | x k ) = hom( F − x, H ◦ ( ℓ )) if k ∈ { q +1 , . . . , q + ℓ } .So hom( F, H ◦ ( ℓ )) = q + ℓ X i =1 hom( F, H ◦ ( ℓ ) | x i )= ℓ hom( F − x, H ◦ ( ℓ )) + q X i =1 X j ∼ i hom( F − x, H ◦ ( ℓ ) | y j )= ℓ hom( F − x, H ◦ ( ℓ )) + q + ℓ X j = q +1 q hom( F − x, H ◦ ( ℓ ) | y j )+ q X j =1 d H ( j ) hom( F − x, H ◦ ( ℓ ) | y j )= ℓ hom( F − x, H ◦ ( ℓ )) + q + ℓ X j = q +1 ( q − ∆) hom( F − x, H ◦ ( ℓ ) | y j )+∆ q + ℓ X j =1 hom( F − x, H ◦ ( ℓ ) | y j )= ( q − ∆) hom( F − x − y, H ◦ ( ℓ )) + (∆ + ℓ ) hom( F − x, H ◦ ( ℓ )) . If F = P n then F − x = P n − and F − x − y = P n − . Therefore, by induction, we havehom( F, H ◦ ( ℓ )) = ( q − ∆) hom( F − x − y, H ◦ ( ℓ )) + (∆ + ℓ ) hom( F − x, H ◦ ( ℓ )) ≥ ( q − ∆) hom( P n − , H ◦ ( ℓ )) + (∆ + ℓ ) hom( P n − , H ◦ ( ℓ ))= hom( P n , H ◦ ( ℓ )) . When can we achieve equality? If H ◦ ( ℓ ) is the complete looped graph, then we haveequality for all F . So we may assume that H ◦ ( ℓ ) is not a complete looped graph, and thatin particular there are i, j ∈ V ( H ◦ ( ℓ )) with i j . Now consider an F with more than onecomponent, and let u and v be vertices of F in different components. Using the looped domi-nating vertices of H ◦ ( ℓ ) we may construct an H ◦ ( ℓ )-coloring of F in which u is colored i and v is colored j . This is not a valid H ◦ ( ℓ )-coloring of the forest obtained from F by adding theedge uv . It follows that there is a tree T on n vertices with hom( F, H ◦ ( ℓ )) > hom( T, H ◦ ( ℓ )).The proof of the Hoffman-London inequality (hom( K ,n − , H ) ≥ hom( P n , H )) given in [17]in fact shows strict inequality for H ◦ ( ℓ ) that is not a complete looped graph; since strictinequality holds for the base cases n ≤
4, the inductive proof therefore gives strict inequalityunless F − x = P n − and F − x − y = P n − , which implies F = P n .5 Proof of Theorem 1.3
First, notice that for ∆-regular H we have hom( T, H ) = | V ( H ) | ∆ n − for all T ∈ T ( n ), ascan be seen by fixing the color on one vertex and iteratively coloring away from that vertex.Therefore the goal for the remainder of this section is to show that for non-regular H thereis a constant c H such that if n ≥ c H and T ∈ T ( n ) thenhom( T, H ) ≤ hom( K ,n − , H ) , with equality if and only if T = K ,n − . Recall that a loop in H will count once toward thedegree of a vertex v ∈ V ( H ) and ∆ denotes the maximum degree of H . If H has components H , . . . , H s then hom( G, H ) = hom(
G, H ) + · · · + hom( G, H s ), so we may assume that H isconnected.Let V =∆ ⊂ V ( H ) be the set of vertices in H with degree ∆. By coloring the center ofthe star with a color from V =∆ , we havehom( K ,n − , H ) ≥ | V =∆ | ∆ n − . We will show that if n ≥ c H and T = K ,n − , then hom( T, H ) < | V =∆ | ∆ n − . We use thefollowing lemma, which will also be needed in the proof of Theorem 1.4. Lemma 3.1.
For non-regular H there exists a constant ℓ H such that if k ≥ ℓ H , then hom( P k , H ) < ∆ k − .Proof. Let A denote the adjacency matrix of H , which is a symmetric non-negative matrix.Since an H -coloring of P k is exactly a walk of length k through H , we havehom( P k , H ) = X i,j ( A k ) ij ≤ | V ( H ) | max i,j ( A k ) ij . (3)By the Perron-Frobenius theorem (see e.g. [15, Theorem 1.5]), the largest eigenvalue λ of A has a strictly positive eigenvector x , for which it holds that A k x = λ k x . Considering therow of A k containing max i,j (cid:0) A k (cid:1) ij , it follows that max i,j (cid:0) A k (cid:1) ij ≤ c λ k for some constant c .Combining this with (3) we find that hom( P k , H ) ≤ | V ( H ) | c λ k . Since H is not regular,we have λ < ∆, and so the lemma follows.We use Lemma 3.1 to show that any tree T containing P k as a subgraph, for k ≥ ℓ H , hashom( T, H ) < | V =∆ | ∆ n − . Indeed, by first coloring the path and then iteratively coloring therest of the tree we obtain hom( T, H ) < ∆ k − ∆ n − k < | V =∆ | ∆ n − . (The bound hom( P k , H ) < ∆ k − will be necessary for the proof of Theorem 1.4; here theweaker bound hom( P k , H ) < | V =∆ | ∆ k − would suffice.)So suppose that T does not contain a path of length ℓ H . Notice that for n > c +1 thereare at most 1 + n / ( c +1) + n / ( c +1) + · · · + n c/ ( c +1) < n c and each vertex having degree at most n / ( c +1) .Because of this, there exists a constant c > ℓ H , but is independent of n ) and a vertex v ∈ T with d ( v ) ≥ n c . Since T = K ,n − we have d ( v ) < n − w of v is adjacent to a vertex in V ( T ) \ ( { v } ∪ N ( v )). As H is connected and notregular, there is an i ∈ V ( H ) with d H ( i ) = ∆ and d H ( j ) ≤ ∆ − j of i .The number of H -colorings of T that don’t color v with a vertex of degree ∆ is at most | V ( H ) | (∆ − n c ∆ n − − n c ≤ | V ( H ) | e − n c / ∆ ∆ n − . (4)The number of H -colorings that color v with a vertex of degree ∆ different from i is at most( | V =∆ | − n − , (5)and the number of H -colorings that color v with i and w with a color different from j is atmost (∆ − n − = (cid:18) − (cid:19) ∆ n − . (6)Finally, the number of H -colorings that color v with i and w with j is at most(∆ − n − = (cid:18) − (cid:19) ∆ n − , (7)since the degree of j is at most ∆ − w is adjacent to a vertex in V ( T ) \ ( { v } ∪ N ( v )),which can be colored with one of the at most ∆ − j . Combining (4), (5), (6),and (7) we obtainhom( T, H ) ≤ (cid:18) | V =∆ | − + | V ( H ) | e − n c / ∆ (cid:19) ∆ n − < | V =∆ | ∆ n − , the final inequality valid as long as n ≥ c H . Here we build on the approach used in the proof of Theorem 1.3 to tackle the family of n -vertex 2-connected graphs C ( n ).In order to proceed, we will need a structural characterization of 2-connected graphs. Inthe definition below we abuse standard notation a little, by allowing the endpoints of a pathto perhaps coincide. Definition 4.1. An ear on a graph G is a path whose endpoints are vertices of G , butwhich otherwise is vertex-disjoint from G . An ear is an open ear if the endpoints of the pathare distinct. An (open) ear decomposition of a graph G is a partition of the edge set of G into parts Q , Q , . . . Q ℓ such that Q is a cycle and Q i for 1 ≤ i ≤ ℓ is an (open) ear on Q ∪ · · · ∪ Q i − . Theorem 4.2 (Whitney [18]) . A graph is -connected if and only if it admits an open eardecomposition. H -colorings, we willassume in the proof of Theorem 1.4 that we are working with a minimally -connected graph G , meaning that G is 2-connected and for any edge e we have that G − e is not 2-connected.A characterization of these graphs is the following, which can be found in [4]. Theorem 4.3. A -connected graph G is minimally -connected if and only if no cycle in G has a chord. Corollary 4.4.
Let G be a minimally -connected graph. There is an open ear decomposition Q , . . . , Q ℓ of G , and a c satisfying ≤ c ≤ ℓ , with the property that each of Q up to Q c arepaths on at least four vertices, and each of Q c +1 through Q ℓ are paths on three vertices withendpoints in ∪ ck =1 Q k .Proof. Let Q , . . . , Q ℓ be an open ear decomposition of G . If any of the Q k , 1 ≤ k ≤ ℓ , is apath on two vertices, its removal leads to an open ear decomposition of a proper spanningsubgraph of G , contradicting the minimality of G . So we may assume that each of Q through Q ℓ is a path on at least three vertices. We claim that if Q i is a path on exactlythree vertices, then the degree 2 vertex in Q i also has degree 2 in G ; from this the corollaryeasily follows.To verify the claim, let Q i be on vertices x, a and y , with a the vertex of degree 2, andsuppose, for a contradiction, that there is some Q j , j > i , that has endpoints a and z (thelatter of which may be one of x , y ). In ∪ i − k =1 Q k there is a cycle C containing both x and y ,and in ∪ j − k =1 Q k there is a path P from z to C that intersects C only once. Now consider thecycle C ′ that starts at a , follows Q j to z , follows P to C , follows C until it has met both x and y (meeting y second, without loss of generality), and finishes along the edge ya . Theedge xa is a chord of this cycle, giving us the desired contradiction.The next lemma follows from results that appear in [5], specifically Lemma 5.3 and theproof of Corollary 5.4 of that reference. Lemma 4.5.
Suppose H is not K ∆ , ∆ or K ◦ ∆ (the complete looped graph). Then for any twovertices i , j of H and for k ≥ there are at most (∆ − k − H -colorings of P k that mapthe initial vertex of the path to i and the terminal vertex to j .Remark. Corollary 5.4 in [5] gives a bound of ∆ k − for a smaller class of H , which is simplyfor convenience. The proof given actually delivers a bound of (∆ − k − for all H except K ∆ , ∆ and K ◦ ∆ . Proof of Theorem 1.4.
We begin with the proof for a non-regular connected graph H , andthen consider regular H and describe the necessary modifications needed in the proof.Let H be non-regular and connected. We will first show that for all sufficiently large n and all minimally 2-connected graphs G that are different from K ,n − we have hom( G, H ) < hom( K ,n − , H ), and we will then address the cases of equality when G is allowed to be notminimal. An easy lower bound on the number of H -colorings of K ,n − ishom( K ,n − , H ) ≥ s ( H )∆ n − ≥ ∆ n − , where s ( H ) is the number of ordered pairs ( i, j ) of (not necessarily distinct) vertices in V ( H )with the property that | N ( i ) ∩ N ( j ) | = ∆. For the first inequality, consider coloring the8ertices in the partition class of size 2 of K ,n − with colors i and j , and for the second notethat the pair ( i, i ), where i is any vertex of degree ∆, is counted by s ( H ).Suppose that G contains a path on k ≥ ℓ H vertices (with ℓ H as given by Lemma 3.1). Byfirst coloring this path and then iteratively coloring the remaining vertices we have (usingLemma 3.1) hom( G, H ) < ∆ k − ∆ n − k = ∆ n − . We may therefore assume that G contains nopath on k ≥ ℓ H vertices.Consider now an open ear decomposition of G satisfying the conclusions of Corollary 4.4.Since G contains no path on ℓ H vertices, we have that there is some constant (independentof n ) that bounds the lengths of each of the Q i , and so ℓ , the number of open ears inthe decomposition, satisfies ℓ = Ω( n ). We now show that c , the number of paths in theopen ear decomposition that have at least four vertices, may be taken to be at most aconstant (independent of n ). Coloring Q first, then coloring each of Q through Q c , andthen iteratively coloring the rest of G , Lemma 4.5 yieldshom( G, H ) ≤ | V ( H ) | ∆ | Q |− (∆ − c ∆( P ci =1 | Q i | ) − c ∆ ℓ − c ≤ | V ( H ) | ∆ (cid:18) − (cid:19) c ∆ n (noting that n = | Q | + (( P ci =1 | Q i | ) − c ) + ( ℓ − c )). Unless c is a constant, this quantityfalls below the trivial lower bound on hom( K ,n − , H ) for all sufficiently large n .Now since there are only constantly many vertices in the graph G ′ with open ear decom-position Q , . . . , Q c , and G is obtained by adding the remaining vertices to G ′ , each joinedto exactly two vertices of G ′ , it follows by the pigeonhole principle that for some constant c ′ there is a pair of vertices w , w in G with at least c ′ n common neighbors, all among themiddle vertices of the Q i for i ≥ c + 1. (Notice that this makes G “close” to K ,n − in thesame sense that in the proof of Theorem 1.3 a tree with a vertex of degree Ω( n ) was “close”to K ,n − .)We count the number of H -colorings of G by first considering those in which w is colored i and w is colored j , for some pair i, j ∈ V ( H ) with | N ( i ) ∩ N ( j ) | < ∆. There are at most | V ( H ) | (∆ − c ′ n ∆ n − − c ′ n (8)such H -colorings. Next, we count the number of H -colorings of G in which w is colored i and w is colored j , for some pair i, j ∈ V ( H ) with | N ( i ) ∩ N ( j ) | = ∆. We argue that in G ′ (the graph with open ear decomposition Q , . . . , Q c ) there must be a path P on at least4 vertices with endpoints w and w . To see this, note that since G = K ,n − there mustbe some vertex v ∈ G ′ so that v = w , v = w , and v / ∈ N ( w ) ∩ N ( w ). Choose a cyclein G ′ containing v and w , and find a path from w to that cycle (the path will be trivial if w is on the cycle). From this structure we find such a path P , and it must have at least 4vertices as v / ∈ N ( w ) ∩ N ( w ).Coloring G by first coloring w and w , then the vertices of P , and finally the rest of thegraph, we find by Lemma 4.5 that the number of H -colorings of G in which w is colored i and w is colored j , for some pair i, j ∈ V ( H ) with | N ( i ) ∩ N ( j ) | = ∆ is at most s ( H ) (cid:18) − (cid:19) ∆ n − . (9)9ombining (8) and (9) we havehom( G, H ) ≤ | V ( H ) | ∆ n − (cid:18) ∆ − (cid:19) c ′ n + s ( H ) (cid:18) − (cid:19) ∆ n − < s ( H )∆ n − ≤ hom( K ,n − , H ) , with the strict inequality valid for all sufficiently large n .We have shown that for non-regular connected H , K ,n − is the unique minimally 2-connected n -vertex graph maximizing the number of H -colorings. We now complete theproof of Theorem 1.4 for non-regular connected H by showing that if G is an n -vertex2-connected graph that is not minimally 2-connected then hom( G, H ) < hom( K ,n − , H ).Suppose for the sake of contradiction that hom( G, H ) = hom( K ,n − , H ). Since deletingedges from G cannot decrease the count of H -colorings, the uniqueness of K ,n − amongminimally 2-connected graphs shows that K ,n − is the only minimally 2-connected graphthat is a subgraph of G , and in particular we may assume that G is obtained from K ,n − by adding a single edge, necessarily inside one partition class of K ,n − . Since K ,n − is asubgraph of G , all H -colorings of G are H -colorings of K ,n − .Suppose that i and j are distinct adjacent vertices of H . The H -coloring of K ,n − thatmaps one partition class to i and the other to j must also be an H -coloring of G , whichimplies that both i and j must be looped. By similar reasoning, if i and j are adjacent in H , and also j and k , then i and k must be adjacent. It follows that H is a disjoint union offully looped complete graphs, and so if connected must be a single complete looped graph(and therefore is regular), which is a contradiction.Now we turn to regular connected H satisfying s ( H ) ≥ + 1 (if H is loopless andbipartite) or s ( H ) ≥ ∆ + 1 (otherwise). For these H , Lemma 3.1 does not apply. We willargue, however, that there is still an ℓ H such that if 2-connected n -vertex G has an open eardecomposition in which any of the added paths Q i , i ≥
1, has at least ℓ H vertices then G hasfewer H -colorings than K ,n − . Once we have established this, the proof proceeds exactly asin the non-regular case.We begin by establishing that G has no long cycles; we will use the easy fact that thenumber of H -colorings of a k -cycle is the sum of the k th powers of the eigenvalues of theadjacency matrix of H . By the Perron-Frobenius theorem there is exactly one eigenvalueof the adjacency matrix of H equal to ∆, and a second one equal to − ∆ if and only if H is loopless and bipartite, with the remaining eigenvalues having absolute value strictly lessthan ∆. It follows that the number of H -colorings of a k -cycle is, for large enough k , lessthan (2 + )∆ k (if H is loopless and bipartite) or (1 + )∆ k (otherwise). If G has sucha cycle then by coloring the cycle first and then coloring the remaining vertices sequentially,bounding the number of options for the color of each vertex by ∆, we get that the numberof H -colorings of G falls below the trivial lower bound on hom( K ,n − , H ) of s ( H )∆ n − .Now if a 2-connected G has an open ear decomposition with an added path Q i , i ≥
1, onat least ℓ H vertices, then it has a cycle of length at least ℓ H , since the endpoints of Q i arejoined by a path in Q ∪ . . . ∪ Q i − . 10 Proof of Theorem 1.5
In this section we show that for all q ≥ n ≥
3, among n -vertex 2-edge-connectedgraphs, and therefore among the subfamily of n -vertex 2-connected graphs, the cycle C n uniquely (up to one small exception) admits the greatest number of proper q -colorings (thatis, K q -colorings). The result is trivial for n = 3 and easily verified directly for n = 4, sothroughout we assume n ≥ Theorem 5.1 (Robbins [14]) . A graph is -edge-connected if and only if it admits an eardecomposition. Notice that we are trying to show that C n , which is the unique graph constructed witha trivial ear decomposition Q = C n , maximizes hom( G, K q ). To do this, we aim to showthat almost any time a 2-edge-connected graph contains a cycle and a path that meetsthe cycle only at the endpoints (which is what results after the first added ear in an eardecomposition for G = C n ), we can produce an upper bound on hom( G, K q ) which is smallerthan hom( C n , K q ); we will then deal with the exceptional cases by hand. The proof willinitially consider an arbitrary fixed cycle and a path joined to that cycle at its endpoints in G (but independent of a particular ear decomposition of G ), and will begin by producing anupper bound on hom( G, K q ) based on the lengths of the path and cycle.Let G = C n be an n -vertex 2-edge-connected graph, and let C be a non-Hamiltoniancycle in G of length ℓ . Since ℓ < n and G is 2-edge-connected, there is a path P on at least3 vertices that meets C only at its endpoints.Suppose first that P contains m + 2 vertices, where m ≥ m vertices on P outside of the vertices of C ). Color G by first coloring C , then P , and then the rest of thegraph. Using hom( C ℓ , K q ) = ( q − ℓ + ( − ℓ ( q −
1) and Lemma 4.5, we havehom(
G, K q ) ≤ (cid:0) ( q − ℓ + ( − ℓ ( q − (cid:1) (cid:2)(cid:0) ( q − − (cid:1) ( q − m − (cid:3) ( q − n − m − ℓ = ( q − n − ( q − n − + ( − ℓ ( q − n − ℓ +1 − ( − ℓ ( q − n − ℓ − ≤ ( q − n − ( q − n − + ( q − n − ℓ +1 − ( q − n − ℓ − ≤ hom( C n , K q ) . Since the last inequality is strict for ℓ > ℓ , the chain of inequalities is strict for all ℓ ≥
3. In other words, if G containsany cycle C and a path P on at least 4 vertices that meets C only at its endpoints, thenhom( G, K q ) < hom( C n , K q ).Suppose next that all the paths on at least three vertices that meet C only at theirendpoints have exactly three vertices. If ℓ ≥
5, or if ℓ = 4 and there is such a path that joinstwo adjacent vertices of the cycle, then using just the vertices and edges of C and one suchpath P we can easily find a new cycle C ′ and a path P ′ on at least four vertices that onlymeets C ′ at it endpoints, and so hom( G, K q ) < hom( C n , K q ) as before.It remains to find an upper bound for those 2-edge-connected graphs (apart from C n )that do not have any of the following: 11a) a cycle of any length with a path on at least 4 vertices that only meets the cycle at itsendpoints;(b) a cycle of length at least 5 and a path on 3 vertices that only meets the cycle at itsendpoints; or(c) a cycle of length 4 with a path of length 3 that only meets the cycle at its endpoints,and those endpoints are two adjacent vertices of the cycle.Recall that we are assuming n ≥
5. For the remaining graphs, notice that if Q is a cycleof length 3, then Q ∪ Q , where Q is a path on three vertices (by (a) above), must containa cycle of length 4. In fact, in this case this cycle will use vertices from both Q and Q .Therefore, we need not separately analyze the situation where C is a cycle of length 3 amongthe remaining graphs. In particular, this means that there is one final case to consider: C has length ℓ = 4, and P is a path on 3 vertices that only meets C at its endpoints, and theendpoints of P are non-adjacent vertices of C . In this case, C and P form a copy of K , ,and we color this copy of K , first and then the rest of G . This giveshom( G, K q ) ≤ (cid:0) q ( q − + q ( q − q − (cid:1) ( q − n − . For n ≥
6, some algebra shows that the right-hand side above is strictly less than hom( C n , K q ),and also for n = 5 and q >
3. For n = 5 and q = 3 we have equality, and so in thiscase hom( G, K ) ≤ hom( C , K ) with equality only if G has the same number of proper3-colorings as K , and has K , as a subgraph; this can only happen if G = K , (using asimilar argument as the one given in the proof of the cases of equality for Theorem 1.4). In this section, we highlight a few questions related to the work in this article.
Question 6.1.
Which graphs H have the property that the path on n vertices is the tree thatminimizes the number of H -colorings? Since the star maximizes the number of H -colorings among trees, it is natural to repeatthe maximization question among trees with some prescribed bound on the maximum degree.Some results (for strongly biregular H ) are obtained in [13]. Question 6.2.
For a fixed non-regular H and positive integer ∆ , which tree on n verticeswith maximum degree ∆ has the most H -colorings? There are still open questions about regular H in the family of 2-connected graphs. Inparticular, we have seen that K ,n − and C n are 2-connected graphs that have the most H -colorings for some H . Question 6.3.
What are the necessary and sufficient conditions on H so that K ,n − is theunique 2-connected graph that has the most H -colorings? uestion 6.4. Are C n and K ,n − the only -connected graphs that uniquely maximize thenumber of H -colorings for some H ? Finally, it would be natural to try to extend these results to k -connected graphs and to k -edge-connected graphs. Question 6.5.
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