Ajit A. Diwan

Efficient inference with cardinality-based clique potentials

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 re-parameterization (TRW) and graph-cut algorithms.

Decomposing graphs with girth at least five under degree constraints

A multigraph M with maximum degree Δ(M) is called critical, if the chromatic index χ2(M) > Δ(M) and χ2(M - e) = χ2(M) - 1 for each edge e of M. The weak critical graph conjecture [1, 7] claims that there exists a constant c > 0 such that every critical multigraph M with at most c · Δ(M) vertices has odd order. We disprove this conjecture by constructing critical multigraphs of order 20 with maximum degree k for all k e 5.

Balanced group-labeled graphs

A group-labeled graph is a graph whose vertices and edges have been assigned labels from some abelian group. The weight of a subgraph of a group-labeled graph is the sum of the labels of the vertices and edges in the subgraph. A group-labeled graph is said to be balanced if the weight of every cycle in the graph is zero. We give a characterization of balanced group-labeled graphs that generalizes the known characterizations of balanced signed graphs and consistent marked graphs. We count the number of distinct balanced labellings of a graph by a finite abelian group @C and show that this number depends only on the order of @C and not its structure. We show that all balanced labellings of a graph can be obtained from the all-zero labeling using simple operations. Finally, we give a new constructive characterization of consistent marked graphs and markable graphs, that is, graphs that have a consistent marking with at least one negative vertex.

Disconnected 2-Factors in Planar Cubic Bridgeless Graphs

We prove that every planar cubic bridgeless graph with at least six vertices has a disconnected 2-factor, that is a 2-factor which is not a Hamilton cycle. This is in contrast to the fact that there exist arbitrarily large cubic bridgeless graphs in which every 2-factor is a Hamilton cycle.

Cycles of even lengths modulo k

Thomassen [J Graph Theory 7 (1983), 261–271] conjectured that for all positive integers k and m, every graph of minimum degree at least 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 at least 2k−1, which improves the previously known bound of 3k−2. We also show that Thomassens conjecture is true for m = 2.

P3→-decomposition of directed graphs

Abstract A P 3 → -decomposition of a directed graph D is a partition of the arcs of D into directed paths of length 2. We characterize symmetric digraphs that do not admit a P 3 → -decomposition. We show that the only 2-regular, connected directed graphs that do not admit a P 3 → -decomposition are obtained from undirected odd cycles by replacing each edge by two oppositely directed arcs. In both cases, we give a linear-time algorithm to find a P 3 → -decomposition, if it exists.

Circumference, chromatic number and online coloring

Erdős conjectured that if G is a triangle free graph of chromatic number at least k≥3, then it contains an odd cycle of length at least k2−o(1) [13,15]. Nothing better than a linear bound ([3], Problem 5.1.55 in [16]) was so far known. We make progress on this conjecture by showing that G contains an odd cycle of length at least Ω(k log logk). Erdős’ conjecture is known to hold for graphs with girth at least five. We show that if a graph with girth four is C5 free, then Erdős’ conjecture holds. When the number of vertices is not too large we can prove better bounds on χ. We also give bounds on the chromatic number of graphs with at most r cycles of length 1 mod k, or at most s cycles of length 2 mod k, or no cycles of length 3 mod k. Our techniques essentially consist of using a depth first search tree to decompose the graph into ordered paths, which are then fed to an online coloring algorithm. Using this technique we give simple proofs of some old results, and also obtain several other results. We also obtain a lower bound on the number of colors which an online coloring algorithm needs to use to color triangle free graphs.

Degree conditions for forests in graphs

If H is any forest of order n with m edges, then any graph G of order >=n with d(u)+d(v)>=2m-1 for any two non-adjacent vertices u,v contains H.

Plane triangulations are 6-partitionable

Given a graph G = (V,E) and k positive integers n1,n2 ..... nk such that Σki = 1 ni = |V|,we wish to find a partition P1,P2,..... Pk of the vertex set V such that |Pi| = ni and Pi induces a connected subgraph of G for all i, 1 ≤ i ≤ k. Such a partition is called a k-partition of G. A graph G with n vertices is said to be k-partitionable if there exists a k-partition of G for any partition of n into k parts. Lovasz (Acta Math. Acad. Sci. Hungar. 30 (1977) 241) showed that k-connected graphs are k-partitionable. In this paper we prove that plane triangulations are 6-partitionable. This result is the best possible as there exist plane triangulations which are not 7-partitionable.

On Colouring Point Visibility Graphs

In this paper we show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4.