Abstract
An important theorem due to Truemper characterizes the graphs whose edges can be labelled so that all chordless cycles have prescribed parities. This theorem has since proved an essential tool in the study of balanced matrices, graphs with no even length chordless cycle and graphs with no odd length chordless cycle of length greater than 3. In this paper we prove this theorem in a novel and elementary way and we derive some of its consequences. In particular, we show how to obtain Tutte’s characterization of regular matrices.
Supported in part by a grant from Gruppo Nazionale Delle Ricerche-CNR.
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References
P. Camion. Caractérisation des matrices totalement unimodulaires. Cahiers Centre Études Rech. Op., 5:181–190, 1963.
M. Conforti, G. Cornuéjols, A. Kapoor, and K. Vušsković. Balanced 0, ±1 matrices, Parts I–II. 1994. Submitted for publication.
M. Conforti, G. Cornuéjols, A. Kapoor, and K. Vušković. Even-hole-free graphs, Parts I–II. Preprints, Carnegie Mellon University, 1997.
M. Conforti, G. Cornuéjols, A. Kapoor, and K. Vušković. Even and odd holes in cap-free graphs. 1996. Submitted for publication.
M. Conforti, G. Cornuéjols, A. Kapoor, and K. Vušković. Universally signable graphs. Combinatorica, 17(1):67–77, 1997.
M. Conforti, G. Cornuéjols, A. Kapoor, and K. Vušković. A Mickey-Mouse decomposition theorem. In Balas and Clausen, editors, Proceedings of 4th IPCO Conference, Springer Verlag, 1995.
M. Conforti, G. Cornuéjols, A. Kapoor, M. R. Rao, and K. Vušković. Balanced matrices. Proceedings of the XV International Symposium on Mathematical Programming, University of Michigan Press, 1994.
M. Conforti, G. Cornuéjols, and M. R. Rao. Decomposition of balanced 0,1 matrices, Parts I–VII. 1991. Submitted for publication.
M. Conforti, G. Cornuéjols, and K. Vušsković. Balanced cycles and holes in bipartite graphs. 1993. Submitted for publication.
A. M. H. Gerards. A short proof of Tutte’s characterization of totally unimodular matrices. Linear Algebra and its Applications, 14:207–212, 1989.
A. Hajnal and T. Suryani. Uber die auflosung von graphen vollstandiger teilgraphen. Ann. Univ. Sc. Budapest. Eotvos Sect. Math., 1, 1958.
P. Seymour. Decomposition of regular matroids. Journal of Combinatorial Theory B, 28:305–359, 1980.
K. Truemper. On balanced matrices and Tutte’s characterization of regular matroids. Working paper, University of Texas at Dallas, 1978.
K. Truemper. Alpha-balanced graphs and matrices and GF(3)-representability of matroids. Journal of Combinatorial Theory B, 32:112–139, 1982.
K. Truemper. A decomposition theory of matroids V. Testing of matrix total unimodularity. Journal of Combinatorial Theory B, 49:241–281, 1990.
W. T. Tutte. A homotopy theorem for matroids I, II. Trans. Amer. Math. Soc. 88:144–174, 1958.
W. T. Tutte. Lectures on matroids. J. Nat. Bur. Standards B, 69:1–47, 1965.
M. Yannakakis. On a class of totally unimodular matrices. Mathematics of Operations Research, 10:280–304, 1985.
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© 1998 Springer-Verlag Berlin Heidelberg
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Conforti, M., Kapoor, A. (1998). A Theorem of Truemper. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds) Integer Programming and Combinatorial Optimization. IPCO 1998. Lecture Notes in Computer Science, vol 1412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-69346-7_5
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DOI: https://doi.org/10.1007/3-540-69346-7_5
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