Sparse Kneser graphs are Hamiltonian

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form $K(2k+2^a,k)$ with $k\geq 3$ and $a\geq 0$ have a Hamilton cycle. We also prove that $K(2k+1,k)$ has at least $2^{2^{k-6}}$ distinct Hamilton cycles for $k\geq 6$. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words

Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and employ deletion-contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure

Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators

Matroids arising from electrical networks

This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main results. First, we exhibit a matroid quotient formed by the dual of a network embedded in a surface with boundary and the dual of the associated Dirichlet matroid. This generalizes an analogous result for graphic matroids of cellularly embedded graphs. Second, we characterize the Bergman fans of Dirichlet matroids as explicit subfans of graphic Bergman fans. In doing so, we generalize the connection between Bergman fans of complete graphs and phylogenetic trees. Third, we use the half-plane property of Dirichlet matroids to prove an interlacing result on the real zeros and poles of the trace of the response matrix. And fourth, we bound the coefficients of the precoloring polynomial of a network by the coefficients of the chromatic polynomial of the underlying graph.Comment: 27 pages, 14 figure