The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems

We show that an effective version of Siegel’s Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the κ ≥ 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. This dichotomy holds even when restricted to planar graphs. A special case of this result is that counting edge κ-colorings is #P-hard over planar 3-regular graphs for κ ≥ 3. In fact, we prove that counting edge κ-colorings is #P-hard over planar r-regular graphs for all κ ≥ r ≥ 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.

Counting Independent Sets and Colorings on Random Regular Bipartite Graphs

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. In the weighted case, for all sufficiently large integers Delta and weight parameters lambda = Omega~ (1/(Delta)), we also obtain an FPTAS on almost every Delta-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all q >= 3 and sufficiently large integers Delta=Delta(q), there is an FPTAS to count the number of q-colorings on almost every Delta-regular bipartite graph

Sidorenko's conjecture, colorings and independent sets

Let $\hom(H,G)$ denote the number of homomorphisms from a graph $H$ to a graph $G$. Sidorenko's conjecture asserts that for any bipartite graph $H$, and a graph $G$ we have $$\hom(H,G)\geq v(G)^{v(H)}\left(\frac{\hom(K_2,G)}{v(G)^2}\right)^{e(H)},$$ where $v(H),v(G)$ and $e(H),e(G)$ denote the number of vertices and edges of the graph $H$ and $G$, respectively. In this paper we prove Sidorenko's conjecture for certain special graphs $G$: for the complete graph $K_q$ on $q$ vertices, for a $K_2$ with a loop added at one of the end vertices, and for a path on $3$ vertices with a loop added at each vertex. These cases correspond to counting colorings, independent sets and Widom-Rowlinson colorings of a graph $H$. For instance, for a bipartite graph $H$ the number of $q$-colorings $\textrm{ch}(H,q)$ satisfies $$\textrm{ch}(H,q)\geq q^{v(H)}\left(\frac{q-1}{q}\right)^{e(H)}.$$ In fact, we will prove that in the last two cases (independent sets and Widom-Rowlinson colorings) the graph $H$ does not need to be bipartite. In all cases, we first prove a certain correlation inequality which implies Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande

Manifolds associated with (Z2)n(Z_2)^n-colored regular graphs

In this article we describe a canonical way to expand a certain kind of $(\mathbb Z_2)^{n+1}$-colored regular graphs into closed $n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed combinatorial $n$-manifold can be obtained in this way. When $n\leq 3$, we give simple equivalent conditions for a colored graph to admit an expansion. In addition, we show that if a $(\mathbb Z_2)^{n+1}$-colored regular graph admits an $n$-skeletal expansion, then it is realizable as the moment graph of an $(n+1)$-dimensional closed $(\mathbb Z_2)^{n+1}$-manifold.Comment: 20 pages with 9 figures, in AMS-LaTex, v4 added a new section on reconstructing a space with a $(Z_2)^n$-action for which its moment graph is a given colored grap