A graph G is pseudo 2-factor isomorphic if the parity of the number of circuits in a 2-factor is the same for all 2-factors of G. We prove that there exist no pseudo 2-factor isomorphic k-regular bipartite graphs for k 4. We also propose a characterization for 3-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs and obtain some partial results towards our conjecture.

Many collective labeling tasks require inference on graphical models where the clique potentials depend only on the number of nodes that get a particular label. We design efficient inference algorithms for various families of such potentials.Our algorithms are exact for arbitrary cardinality-based clique potentials on binary labels and for max-like and majority-like clique potentials on multiple labels. Moving towards more complex potentials, we show that inference becomes NP-hard even on cliques with homogeneous Potts potentials. We present a 13 15 -approximation algorithm with runtime sub-quadratic in the clique size. In contrast, the best known previous guarantee for graphs with Potts potentials is only 0.5.We perform empirical comparisons on real and synthetic data, and show that our proposed methods are an order of magnitude faster than the well-known Tree-based reparameterization (TRW) and graph-cut algorithms.

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We prove that the vertex set of a simple graph with minimum degree at least s + t − 1 and girth at least 5 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively, where s, t are positive integers ≥ 2.In [2], M. Stiebitz showed that the vertex set of a graph with minimum degree at least s + t + 1 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively. In [1], A. Kaneko showed that this result holds for triangle-free graphs with minimum degree at least s + t. In this article, we extend this result, using essentially similar ideas, and show that the bound can be further improved to s + t − 1 for graphs with girth at least 5, where s, t ≥ 2. More precisely, we prove the following.Theorem. The vertex set of any simple graph with minimum degree at least s + t − 1 and girth at least 5 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively, where s, t are positive integers ≥ 2.

Thomassen [9] conjectured that for all natural numbers k > 0 and m, every graph of minimum degree k + 1 contains a cycle of length congruent to 2m modulo k. We prove that this is true for k ≥ 2 if the minimum degree is 2k − 1, which improves the previously known bound of 3k − 2. We also show that Thomassen's conjecture is true for m = 2.