Covers, orientations and factors

Given a graph $G$ with only even degrees let $\varepsilon(G)$ be the number of Eulerian orientations and let $h(G)$ denote the number of half graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)/2$ for each vertex $v$. Recently, M. Borb\'enyi and P. Csikv\'ari proved that $\varepsilon(G)\geq h(G)$ holds true for all Eulerian graphs with equality if and and only if $G$ is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a $2$-cover of a graph $G$.


Introduction
Given an Eulerian graph G. We call a graph Eulerian if each vertex has even degree. In the literature it is often assumed that an Eulerian graph is also connected, but we will not require connectedness in this paper. Let ε(G) be the number of Eulerian orientations, that is, the orientations where each vertex has in-degree equal to the out-degree. Counting Eulerian orientations has triggered considerable interest both in combinatorics, computer science and statistical physics. Probably, the best known result is due to Lieb [10] who determined the asymptotic number of Eulerian orientations of large grid graphs. Schrijver [14] gave a lower bound on the number of Eulerian orientations in terms of the degree sequence. Welsh [17] observed that for a 4-regular graph the Tutte-polynomial evaluation |T G (0, −2)| is exactly the number of Eulerian orientations since nowhere-zero Z 3 -flows and Eulerian orientations are in one-to-one correspondence for 4-regular graphs. Mihail and Winkler [12] gave an efficient randomized algorithm to sample and approximately count Eulerian orientations.
Let h(G) denote the number of half graphs, that is, subgraphs F such that d F (v) = d G (v)/2 for each vertex v. Note that h(G) > 0 if G is not only Eulerian, but each of its connected component has an even number of edges. This condition is clearly necessary to have a half graph, and also sufficient: every second edge of an Eulerian tour will determine a half graph.
Recently, M. Borbényi and P. Csikvári [4] proved that ε(G) ≥ h(G) holds true for all Eulerian graphs with equality if and and only if G is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a 2-cover of a graph G. In fact, we study the number of orientations and factors of 2-covers with prescribed in-degree and degree, respectively.
1.1. Results. For an edge set A ⊆ E(G) let ε(A) denote the number of Eulerian orientations of the graph (V, A). Similarly, let h(A) denote the number of half graphs of the graph (V, A). Our first result is an identity for ε(G) and h(G). Theorem 1.1. Let G be an Eulerian graph with edge set E. Then Using the identities of Theorem 1.1 we can easily give a new proof of the following theorem of M. Borbényi and P. Csikvári [4]. Theorem 1.2 (M. Borbényi and P. Csikvári [4]). Let G be an Eulerian graph. Then ε(G) ≥ h(G) with equality if and only if G is bipartite.
As we will see it is natural to consider the number of Eulerian orientations and half graphs of 2-covers of an Eulerian graph G.
, then there are no edges between (u, i) and (v, j) for 0 ≤ i, j ≤ k − 1. When k = 2 one can encode the 2-lift H by putting signs on the edges of the graph G: the + sign means that we use the matching ((u, 0), (v, 0)), ((u, 1), (v, 1)) at the edge (u, v), the − sign means that we use the matching ((u, 0), (v, 1)), ((u, 1), (v, 0)) at the edge (u, v). For instance, if we put + signs to every edge, then we simply get G ∪ G as H, and if we put − signs everywhere, then the obtained 2-cover H is simply Graph cover techniques played important roles in the resolution of many open problems. A. Marcus, D. Spielmann and N. Srivastava [11] used graph covers to construct Ramanujan graphs. The idea was suggested by Y. Bilu and N. Linial [3]. Y. Zhao [19] used the bipartite double cover to prove a conjecture of N. Alon [2] and J. Kahn [9] on the number of independent sets. Later he developed his ideas in the paper [20]. N. Ruozzi [13] proved a conjecture of A. Sudderth, E. Wainwright, and M. Willsky [18] on the Bethe approximation of an attractive graphical model by building on an observation due to P. Vontobel [16] connecting graph covers with Bethe approximation. P. Csikvári [6] combined graph covers with graph limit theory to prove the so-called Lower Matching Conjecture of S. Friedland, E. Krop and K. Markström [7]. The properties of random lifts are also widely studied, see for instance the papers [1] and [8].
Theorem 1.4. Let G be an Eulerian graph with edge set E. Then Combining Theorem 1.4 with Theorem 1.2 we get that These inequalities can be generalized as follows. Let H be an arbitrary 2-cover of an Eulerian graph G. Then In fact, even more general statement is true. To spell out this generalization we need the concepts ε r (G) and h r (G).
We will call such an orientation an r-orientation.
Similarly, let h r (G) be the number of subgraphs F of G with degree d F (v) = r v for each vertex v. We will call such a subgraph an r-factor.
In the following statements we can even drop the condition of G being Eulerian.
. Let H be an arbitrary 2-cover of the graph G. Let us denote by r the induced vector of r G on H and G ∪ G. Then . Let H be an arbitrary 2-cover of a graph G. Let us denote by r the induced vector of r G on H and G × K 2 . Then Note that Theorems 1.7 and 1.8 provide yet another proof of Theorem 1.2 by Next we generalize the concept of Eulerian orientations and half graphs. To every edge of G assign either o or s, that is, orientation or subgraph. Then for all edge we have two choices: if we assigned o to the edge, then we need to orient it, so we choose one of the end points and add 1 to it, and add 0 to the other endpoint. (More precisely, we consider the in-degree.) If we assigned s for the edge, then we need to decide whether we put this edge into subgraph or not, so we either add 1 to both endpoints, or add 0 to both endpoints. We will call such a configuration a factorientation. We will call the contribution of the edges to the vertex v the mixed degree of v, that is, it is the sum of the in-degree from oriented edges and the degree coming from the subgraph. After we choose o or s for every edge, we say that a factorientation is balanced, if for every vertex v the mixed degree is d G (v)/2. Let g(G) be the number of balanced factorientations. We can see that this is a generalization of both Eulerian orientations and half graphs, because if we assign o to each edge, then g(G) = ε(G) and if we assign s to each edge, then g(G) = h(G). Theorem 1.9. Let H be a 2-cover of G = (V, E) encoded with + and − signs, such that for each edge the 2-lifts of this edge get the same letter (o or s). For an edge subset B ⊆ E let B denote the graph with vertex set V and edge set B, just for each edge with a minus in it we swap o to s and vice versa. Then Clearly, Theorem 1.1 and 1.4 are corollaries of this theorem. We can also generalize Theorem 1.2 as follows.
In case of Eulerian orientations let S and T be two Eulerian orientations, and let A be the set of edges where the orientations coincide and B = E \ A be the remaining edges. Similarly to the previous discussion S restricted to A and B gives an Eulerian orientation. The rest of the proof is the same.
Proof of Theorem 1.4. The proof is very similar to the proof of Theorem 1.1. Since . Consider a half graph S of G × K 2 . Let π : G × K 2 → G be the natural projection. For k = 0, 1, 2 let A k = {e ∈ E | |π −1 (e) ∩ S| = k}. Since S was a half graph of G × K 2 we get that for each vertex v we have |E v ∩ A 0 | = |E v ∩ A 2 |. In other words, A 0 is a half graph of A 0 ∪ A 2 . Let us orient an edge (u, v) ∈ A 1 from u to v if (u 0 , v 1 ) ∈ S. Since S was a half graph of G × K 2 this gives an Eulerian orientation of the edges of A 1 . Clearly, from the Eulerian orientation of A 1 and the half graph A 0 of A 0 ∪ A 2 we can immediately reconstruct S.
Proof of Theorem 1.2. We prove the statement by induction on the number of edges of G. If the graph has no edge, then the statement is trivial. If it consists of a single cycle of length k together with isolated vertices, then ε(G) = 2 and h(G) = 0 or 2 depending on k being odd or even. So in this case the theorem is true. If G is different from a single cycle together with isolated vertices, then we use induction: Since ε(G) ≥ 1 and the function x 2 − 2x is a monotone increasing function for x ≥ 1 we get that ε(G) ≥ h(G).

Proof of Theorems 1.7 and 1.8
In this section we prove Theorems 1.7 and 1.8. The proofs rely on three observations, one of them is due to A. Schrijver relating the number of r-orientations of a graph G to the number of perfect matchings of a certain bipartite graph G * constructed from G. (In fact, we will slightly modify it, but still attribute it to A. Schrijver.) A similar observation connects the number of r-factors of a graph G to the number of perfect matchings of another graph G * * constructed from G. The last observation is due to P. Csikvári and gives an inequality between the number of perfect matchings of certain 2-covers of a graph G. Lemma 3.1 (A. Schrijver [14]). Let G be a graph, and let G * be the following bipartite graph. On one side of the bipartite graph every vertex corresponds to an edge e ∈ E(G). On the other side of the bipartite graph we take r v copies of each vertex v. Finally, an edge e = (u, v) is adjacent to all copies of u and v. Then There is an 1 to R map from the set of r-orientations to the set of perfect matchings of G * . Namely, if the edges (u, v i ) ∈ E(G) for i = 1, . . . r u are oriented towards u, then we take the union of perfect matchings between e uv i and the r u copies of u to get a perfect matching of G * .
A similar lemma enable us to encode h r via perfect matchings. A qualitative version of this lemma appeared in [15]. Lemma 3.2 (W. Tutte [15]). Let G be a graph, and let G * * be the following graph. For each edge e = (u, v) we introduce two vertices e uv and e vu , and for each vertex v we introduce r v copies of v. Then we connect e uv with e vu , and we also connect the r u copies of u with e vu for each v ∈ N G (u). Then Proof. Let R = v∈V (G) r v !. Again there is an 1 to R map from the set of r-factors to the set of perfect matchings of G * * . Namely, if the edges (u, v i ) ∈ E(G) for i = 1, . . . r u are in the r-factor, then to get a perfect matching of G * * , we take the union of perfect matchings between e v i u and the r u copies of u together with those edges (e xy , e yx ) for which (x, y) ∈ E(G), but is not in the r-factor.
Next we need a lemma that relates perfect matchings with covers. Lemma 3.3 (P. Csikvári [6,5]). Let G be a graph, and let H be an arbitrary 2-cover of G. Then pm(H) ≤ pm(G × K 2 ). In particular, if G is bipartite, then pm(H) ≤ pm(G) 2 .
Sketch of the proof. Let us project the edges of a perfect matching of a 2-cover H to the graph G. The obtained configuration consists of cycles and double-edges, that is, two edges projected to the same edge. (Every degree must be 2 in the obtained configuration.) For such a configuration we can count the number of preimages. Each cycle can be lifted in at most 2 ways since the preimage of one edge determines the preimage of the subsequent edges in the cycle. It may occur though that we cannot close the cycle. (This happens for instance if try to lift a 3-cycle in the union of two 3-cycles.) On the other hand, it is easy to see that on G × K 2 every cycle can be lifted in exactly 2 ways. This means that every configuration has at least as many preimages on G × K 2 than on another 2-cover H.
Proof of Theorem 1.7. Let H be an arbitrary cover of the graph G. Let us construct the bipartite graphs G * and H * of Lemma 3.1. Observe that H * is also a 2-cover of G * , and so by Lemma 3.3 we have pm(G * ) 2 ≥ pm(H * ). Using Lemma 3.1 we have Hence ε r (G) 2 ≥ ε r (H).
Remark 3.4. An interesting application of Theorem 1.7 is the following. Let T n,m be the toroidal grid of size n × m, that is, a grid of size n × m closed in a toroidal way to make it 4-regular. Then ε(T n,m ) ≥ 4 3 3nm/2 . This can can be seen as follows. E.
Proof of Theorem 1.8. Let H be an arbitrary cover of the graph G. Let us construct the bipartite graphs G * * and H * * of Lemma 3.2. Observe that H * * is also a 2-cover of G * * , and (G × K 2 ) * * = G * * × K 2 and so by Lemma 3.3 we have pm((G × K 2 ) * * ) ≥ pm(H * * ).
Using Lemma 3.2 we have Hence h r (G × K 2 ) ≥ h r (H).

general 2-cover
In this section we prove Theorems 1.9 and 1.10.
Proof of Theorem 1.9. Consider a balanced factorientation S of the graph H. Take the natural projection from H to G, and let A be the set of edges for which the projected edges coincide, that is, both edges are oriented in the same way if there is an o on that edge, or they are both or neither in the subgraph if there is an s on that edge. Let B = E \ A. The factorientation S was balanced so for all vertex v the mixed degrees of v 0 and v 1 are both d G (v)/2. This means that after the natural projectionand the doubling of the original edges of the graph-the mixed degree of vertex v is d G (v). If an edge is in A, then it contributes either 0 or 2 to the mixed degree of v, otherwise it contributes 1. Thus there must be equal number of 0's and 2's contributions, which means that if we restrict the graph to A, then we also get a balanced factorientation.
For an edge (u, v) ∈ B if (u, v) has a plus sign on it, then it contributes the same amount to the mixed degree of u and v as it contributed to the mixed degrees of u 0 and v 0 with the edge (u 0 , v 0 ) in S.
If (u, v) has a minus sign on it, then in (u, v) change s to o and vice verse. If it was o before and the orientations were u 0 → v 1 and v 0 → u 1 , then we do not put the edge (u, v) into the subgraph. If it was o before and the orientations were u 1 → v 0 and v 1 → u 0 , then we do put the edge (u, v) into the subgraph. Note that the contribution of these edges to the mixed degree of a vertex u is the same as the contribution of the original edges to the vertex u 0 .
Similarly, if it was a s before, and (u 0 , v 1 ) was in the subgraph, but (u 1 , v 0 ) was not, then orient the edge u from v 0 to u 0 . This way it is again true that the contribution of these edges to the mixed degree of a vertex u is the same as the contribution of the original edges to the vertex u 0 .
So for all (u, v) ∈ B we made sure that the contribution of the orientation or factor to the mixed degree of a vertex u is the same contribution as the original edges to the mixed degree of the vertex u 0 . Finally, observe that since the mixed degree of u 0 and u 1 was the same, and the edges of A contributed the same to the mixed degrees, it is necessary that the mixed degree contributed by the edges of B to the vertex u is exactly d B (u)/2. This means that the constructed factorientation is balanced if we restrict to the edges of B.
Finally, observe that if we get a balanced factorientation of A and E \ A then we can easily get back the balanced factorientation of H. This finishes the proof.
Proof of Theorem 1.10. The proof is practically the same as the proof of Theorem 1.2. We use induction to the number of edges. We have g(G) 2 = g(G ∪ G) = A⊆E g(A)g(E \ A) = A⊆E g(A)g (E \ A) since G ∪ G corresponds to the 2-cover with only + signs. By induction we have g(A) ≤ ε(A) if A = E. Hence g(G) 2 − 2g(G) ≤ ε(G) 2 − 2ε(G) which implies that g(G) ≤ ε(G).

Open problem
We end this paper with an open problem.
Conjecture 5.1. Let G be an Eulerian graph, and let H be a k-cover of G. Then ε(G) k ≥ e(H).