arXiv subjects

Ajit A. Diwan

Publications and source records attributed to Ajit A. Diwan.

9 recordsLinked to original sources

arXiv2026

$θ$-free matching covered graphs: characterization and consequences

The Ear Decomposition Theorem of Lovász & Plummer (1986) implies that every matching covered graph (MCG), except $K_2$ and cycles, contains (at least) one of $θ$ and $K_4$ as a conformal minor. Lovász [Combinatorica 1983] proved the refinement that every nonbipartite MCG contains one of $K_4$ and $\overline{C_6}$. These immediately lead to three problems: characterize (i) $θ$-free graphs, (ii) $K_4$-free graphs and (iii) $\overline{C_6}$-free graphs. Kothari and Murty [JGT 2016] used the tight cut decomposition theory to solve the planar case of (ii) and (iii); the nonplanar cases are open. In contrast, we exploit a seminal result of Edmonds, Lovász and Pulleyblank [Combinatorica 1982] to obtain a structural characterization of $θ$-free graphs that immediately places the corresponding decision problem in P. The Petersen graph plays a key role. We deduce that every $θ$-free graph has at most $2n-2$ edges, and we characterize the tight examples. Despite being sparse, these graphs are not necessarily planar. In the style of Little [JCT-B 1975], we characterize Pfaffian $θ$-free graphs in terms of their forbidden conformal minors. Using the works of Robertson, Seymour and Thomas [Ann. of Math. 1999], and of McCuaig [E-JC 2004], we deduce that the Pfaffian recognition problem is in P for $θ$-free graphs. Deciding whether a cubic graph is 3-edge-colorable is NP-complete; for $θ$-free ones, we provide a characterization of those that are 3-edge-colorable, and deduce that the corresponding decision problem lies in P. McCuaig [JGT 2000] characterized 3-connected bipartite cubic graphs each of whose conformal cycles is of length 2 $\pmod{4}$; the 2-connected case is open. We stumbled upon the serendipitous corollary of our main result that each conformal cycle of a 2-connected cubic graph is of length 0 $\pmod{4}$ if and only if it is $θ$-free.

math.CO↗

arXiv2025

Graphs with large maximum forcing number

For a graph $G$ with order $2n$ and a perfect matching, let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ respectively. Then $0\leq f(G)\leq F(G)\leq n-1$. Liu and Zhang [10] ever proposed a conjecture: $e(G)\geq \frac{n^2}{n-F(G)}$, where $e(G)$ denotes the number of edges of $G$. In this paper we confirm this conjecture and obtain $F(G)\leq n-\frac{n^2}{e(G)}$. If $F(G)=n-1$, Liu and Zhang [9] proved that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along 4-cycles. If $G$ is bipartite and $F(G)\geq n-k$, $1\leq k\leq n-1$, we show that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along even cycles of length at most $2(k+1)$. Finally, we ask whether $f(G)\geq \lceil\frac{n}{k}\rceil-1$ holds for such bipartite graphs $G$, and give positive answers for the cases $k=1,2$. Further we show all minimum forcing numbers of the bipartite graphs $G$ of order $2n$ and with $F(G)=n-2$ form an integer interval $[\lfloor\frac{n}{2}\rfloor, n-2]$.

math.CO↗

arXiv2025

Extremal minimal bipartite matching covered graphs

A connected graph, on four or more vertices, is matching covered (aka 1-extendable) if every edge is present in some perfect matching. An ear decomposition theorem exists for bipartite matching covered graphs due to Hetyei. From the results and proofs of Lovász and Plummer, that rely on Hetyei's theorem, one may deduce that any minimal bipartite matching covered graph has at least $2(m-n+2)$ vertices of degree two (where minimal means that deleting any edge results in a graph that is not matching covered); such a graph is said to be extremal if it attains the stated lower bound. In this paper, we provide a complete characterization of the class of extremal minimal bipartite matching covered graphs. In particular, we prove that every such graph $G$ is obtained from two copies of a tree devoid of degree two vertices, say $T$ and $T'$, by adding edges -- each of which joins a leaf of $T$ with the corresponding leaf of $T'$. Apart from the aforementioned bound, there are four other bounds that appear in, or may be deduced from, the work of Lovász and Plummer. Each of these bounds leads to a notion of extremality. In this paper, we obtain a complete characterization of all of these extremal classes and also establish relationships between them. Two of our characterizations are in the same spirit as the one stated above. For the remaining two extremal classes, we reduce each of them to one of the already characterized extremal classes using standard matching theoretic operations. A connected graph is k-extendable if it has a matching of cardinality $k$ and each such matching extends to a perfect matching. We also discuss bounds proved by Lou (1999) for minimal k-extendable bipartite graphs. We conjecture stronger bounds and provide evidence for our conjectures by constructing tight examples that are straightforward generalizations of the ones that appear in the 1-extendable case.

math.CO↗

arXiv2024

Cycles of weight divisible by $k$

A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order $f(k)$ of a weighted complete directed graph that does not contain a directed cycle of weight divisible by $k$, for an integer $k \ge 2$, was raised by Alon and Krivelevich [J. Graph Theory 98 (2021) 623-629]. They showed that $f(k)$ is $O(k\log k)$ and $f(k) \le 2k-2$ if $k$ is prime. The best bounds known to us are $f(k) \le 2k-2$ for all $k$ and $f(k) < (3k-1)/2$ for prime $k$. It is also known that $f(k) \ge k$ and this is believed to be the correct value. We prove that $f(k) < k+2Ω(k)$, where $Ω(k)$ is the number of prime factors, not necessarily distinct, in the prime factorization of $k$. We also show that any weighted undirected graph of minimum degree $2k-1$ contains a cycle of weight divisible by $k$. This result is proved in the more general setting in which the weights are from a finite abelian group of order $k$, and the cycle has weight equal to the group identity. We conjecture that this holds for undirected graphs with minimum degree $k+1$.

math.CO↗

arXiv2020

Subdivisions of maximal 3-degenerate graphs of order $d+1$ in graphs of minimum degree $d$

We prove that every graph of minimum degree at least $d \ge 1$ contains a subdivision of some maximal 3-degenerate graph of order $d+1$. This generalizes the classic results of Dirac ($d=3$) and Pelikán ($d=4$). We conjecture that for any planar maximal 3-degenerate graph $H$ of order $d+1$ and any graph $G$ of minimum degree at least $d$, $G$ contains a subdivision of $H$. We verify this in the case $H$ is $P_6^3$ and $P_7^3$

math.CO↗

arXiv2017

The minimum forcing number of perfect matchings in the hypercube

Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of $M$. Pachter and Kim [Discrete Math. 190 (1998) 287--294] conjectured that the forcing number of every perfect matching in the $n$-dimensional hypercube is at least $2^{n-2}$, for all $n \ge 2$. Riddle [Discrete Math. 245 (2002) 283-292] proved this for even $n$. We show that the conjecture holds for all $n \ge 2$. The proof is based on simple linear algebra.

math.CO↗

arXiv2011

A sufficient condition for the existence of an anti-directed 2-factor in a directed graph

Let D be a directed graph with vertex set V and order n. An anti-directed hamiltonian cycle H in D is a hamiltonian cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. An anti-directed 2-factor in D is a vertex-disjoint collection of anti-directed cycles in D that span V. It was proved in [3] that if the indegree and the outdegree of each vertex of D is greater than (9/16)n then D contains an anti-directed hamilton cycle. In this paper we prove that given a directed graph D, the problem of determining whether D has an anti-directed 2-factor is NP-complete, and we use a proof technique similar to the one used in [3] to prove that if the indegree and the outdegree of each vertex of D is greater than (24/46)n then D contains an anti-directed 2-factor.

math.CO↗

arXiv2009

Generalized Collective Inference with Symmetric Clique Potentials

Collective graphical models exploit inter-instance associative dependence to output more accurate labelings. However existing models support very limited kind of associativity which restricts accuracy gains. This paper makes two major contributions. First, we propose a general collective inference framework that biases data instances to agree on a set of {\em properties} of their labelings. Agreement is encouraged through symmetric clique potentials. We show that rich properties leads to bigger gains, and present a systematic inference procedure for a large class of such properties. The procedure performs message passing on the cluster graph, where property-aware messages are computed with cluster specific algorithms. This provides an inference-only solution for domain adaptation. Our experiments on bibliographic information extraction illustrate significant test error reduction over unseen domains. Our second major contribution consists of algorithms for computing outgoing messages from clique clusters with symmetric clique potentials. Our algorithms are exact for arbitrary symmetric potentials on binary labels and for max-like and majority-like potentials on multiple labels. For majority potentials, we also provide an efficient Lagrangian Relaxation based algorithm that compares favorably with the exact algorithm. We present a 13/15-approximation algorithm for the NP-hard Potts potential, with runtime sub-quadratic in the clique size. In contrast, the best known previous guarantee for graphs with Potts potentials is only 1/2. We empirically show that our method for Potts potentials is an order of magnitude faster than the best alternatives, and our Lagrangian Relaxation based algorithm for majority potentials beats the best applicable heuristic -- ICM.

cs.AI↗

arXiv2008

Circumference, Chromatic Number and Online Coloring

Erdös conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound (\cite{gyarfas}, Problem 5.1.55 in \cite{West}) was so far known. We make progress on this conjecture by showing that $G$ contains an odd cycle of length at least $O(k\log\log k)$. Erdös' conjecture is known to hold for graphs with girth at least 5. We show that if a girth 4 graph is $C_5$ free, then Erdös' 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 $1\bmod k$, or at most $s$ cycles of length $2\bmod k$, or no cycles of length $3\bmod k$. 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 simpler results. We also obtain a lower bound on the number of colors an online coloring algorithm needs to use on triangle free graphs.

cs.DM↗