Parity results on connected ƒ-factors

AbstractLet G be a connected graph with vertex set V and let d(ν) denote the degree of a vertex νϵV. For ƒ a mapping from V to the positive integers, an ƒ-factor is a spanning subgraph having degree ƒ(ν) at vertex ν. In this paper we extend the parity results of Thomason [2] on Hamiltonian circuits to connected ƒ-factors. (A Hamiltonian circuit is a connected 2-factor.) We show that if ƒ(ν) and d(ν) have opposite parity for all νϵV then for any given subgraph C there is an even number of connected ƒ-factors having C as a cotree.Let ƒ1 and ƒ2 be any mappings from V to the positive integers that partition d, i.e., d(ν) = ƒ1(ν) +ƒ2(ν) for all νϵV. Let C1 and C2 be any pair of edge disjoint subgraphs. We also show in this paper that the number of decompositions of G into a connected ƒ1-factor having C1 as a cotree and a connected ƒ2-factor having C2 as a cotree is even

On the polynomial parity argument complexity of the combinatorial nullstellensatz

The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems. Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM