Matroid Matching Via Mixed Skew-Symmetric Matrices

Tutte associates a V by V skew-symmetric matrix T, having indeterminate entries, with a graph G=(V,E). This matrix, called the Tutte matrix, has rank exactly twice the size of a maximum cardinality matching of G. Thus, to find the size of a maximum matching it suffices to compute the rank of T. We consider the more general problem of computing the rank of T + K where K is a real V by V skew-symmetric matrix. This modest generalization of the matching problem contains the linear matroid matching problem and, more generally, the linear delta-matroid parity problem. We present a tight upper bound on the rank of T + K by decomposing T + K into a sum of matrices whose ranks are easy to compute.

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Correspondence to James Geelen.

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Part of this research was done while the authors visited the Fields Institute in Toronto, Canada. The research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

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Geelen, J., Iwata, S. Matroid Matching Via Mixed Skew-Symmetric Matrices. Combinatorica 25, 187–215 (2005). https://doi.org/10.1007/s00493-005-0013-7

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Mathematics Subject Classification (2000):

  • 05C70