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Graphs with the Fewest Matchings

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L. Keough, A.J. Radcliffe

Combinatorics · Submitted 7 Oct 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that the graph with the fewest matchings is either the lex or the colex graph.

how it is stated · asserted 1.00 read the record →

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arXiv ID1310.1879 (version 1)
Pith IDVIG4GUH6, Pith's permanent record of this paper and its review
CitationA.J. Radcliffe, L. Keough. "Graphs with the Fewest Matchings." arXiv:1310.1879 (2013). https://pith.science/paper/VIG4GUH6
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,551 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.20 · compares with prior results 0.57 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

In recent years there has been increased interest in extremal problems for "counting" parameters of graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. In the same spirit, the Upper Matching Conjecture claims an upper bound on the number of $k$-matchings in a $d$-regular graph. Here we consider both matchings and matchings of fixed sizes in graphs with a given number vertices and edges. We prove that the graph with the fewest matchings is either the lex or the colex graph. Similarly, for fixed $k$, the graph with the fewest $k$-matchings is either the lex or the colex graph. To prove these results we first prove that the lex bipartite graph has the fewest matchings of all sizes among bipartite graphs with fixed part sizes and a given number of edges.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In recent years there has been increased interest in extremal problems for "counting" parameters of graphs. background 1.00 0.03 0.01 asserted 0.99
s1 For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. background 1.00 0.03 0.01 asserted 1.00
s2 In the same spirit, the Upper Matching Conjecture claims an upper bound on the number of $k$-matchings in a $d$-regular graph. background 1.00 0.03 0.02 speculative 0.58
s3 Here we consider both matchings and matchings of fixed sizes in graphs with a given number vertices and edges. contribution 0.89 0.58 0.03 asserted 0.54
s4 We prove that the graph with the fewest matchings is either the lex or the colex graph. result 1.00 0.96 0.80 asserted 1.00
s5 Similarly, for fixed $k$, the graph with the fewest $k$-matchings is either the lex or the colex graph. result 1.00 0.93 0.30 asserted 1.00
s6 To prove these results we first prove that the lex bipartite graph has the fewest matchings of all sizes among bipartite graphs with fixed part sizes and a given number of edges. method 0.65 0.95 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In recent years there has been increased interest in extremal problems for "counting" parameters of graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. In the same spirit, the Upper Matching Conjecture claims an upper bound on the number of $k$-matchings in a $d$-regular graph. Here we consider both matchings and matchings of fixed sizes in graphs with a given number vertices and edges. We prove that the graph with the fewest matchings is either the lex or the colex graph. Similarly, for fixed $k$, the graph with the fewest $k$-matchings is either the lex or the colex graph. To prove these results we first prove that the lex bipartite graph has the fewest matchings of all sizes among bipartite graphs with fixed part sizes and a given number of edges.

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