Discrete MathematicsΒ· 2026Q1
Minimum forcing numbers of perfect matchings of circular and prismatic graphs
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- 2026year
Short summary
This paper establishes that for a bipartite graph G on n vertices with an involutory weighted adjacency matrix over a field of characteristic not 2, the minimum size of a matching uniquely extendable to a perfect matching in G β‘ Cβk is n, for all k β₯ 2.
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Key points
- Introduces the concept of the minimum forcing number f(G) for a graph G with a perfect matching.
- Extends previous results on f(Qd) and f(G β‘ Kβ) to G β‘ Cβk.
- Proves that for a bipartite graph G on n vertices with an involutory weighted adjacency matrix (characteristic β 2), f(G β‘ Cβk) = n for k β₯ 2.
- Shows the method applies to some unbalanced bipartite graphs with specific weighted bi-adjacency matrices.
AI-generated from the title and abstract; the full text is not read.
Abstract
Let G be a graph with a perfect matching. Denote by π β‘ ( πΊ ) the minimum size of a matching in G that is uniquely extendable to a perfect matching in G . Diwan (2019) used linear algebra to prove that for the d -hypercube π π ( π β₯ 2 ) , π β‘ ( π π ) = 2 π β 2 , thus settling a conjecture of Pachter and Kim (1998). Recently, Mohammadian generalized this method to prove a general result: for a bipartite graph G on n vertices, if G admits an involutory weighted adjacency matrix A over a field F , then π β‘ ( πΊ β‘ πΎ 2 ) = π 2 , where β‘ denotes the Cartesian product of two graphs. In this paper we obtain π β‘ ( πΊ β‘ πΆ 2 β’ π ) = π when a bipartite graph G on n vertices admits an involutory weighted adjacency matrix A over a field F of characteristic not 2, for all integers π β₯ 2 . Moreover, we demonstrate that this method can also be applied to some unbalanced bipartite graphs G when G admits a weighted bi-adjacency matrix with orthogonal rows.
The authors' abstract, as published at the source. Discrete Mathematics, 2026 Β· DOI β
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Field: Geometry and Topology
Geometry and TopologyMathematics