Algebraic Multiplicity of the Eigenvalues of a Bipartite Tournament Matrix

We define the deformation multiplicity of a map germ f: (Cn, 0) → (Cp, 0) with respect to a Boardman symbol i of codimension less than or equal to n and establish a geometrical interpretation of this number in terms of the set of Σi points that appear in a generic deformation of f. Moreover, this number is equal to the algebraic multiplicity of f with respect to i when the corresponding associated ring is Cohen-Macaulay. Finally, we study how algebraic multiplicity behaves with weighted homogeneous map germs.

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The algebraic multiplicity

Research Notes in Mathematics Series - Spectral Theory and Nonlinear Functional Analysis ◽

10.1201/9781420035506.ch4 ◽

2001 ◽

Keyword(s):

Algebraic Multiplicity

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Algebraic Multiplicity Through Logarithmic Residues

Algebraic Multiplicity of Eigenvalues of Linear Operators - Operator Theory: Advances and Applications ◽

10.1007/978-3-7643-8401-2_9 ◽

2007 ◽

pp. 225-247

Keyword(s):

Algebraic Multiplicity

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Algebraic Multiplicity Through Transversalization

Algebraic Multiplicity of Eigenvalues of Linear Operators - Operator Theory: Advances and Applications ◽

10.1007/978-3-7643-8401-2_4 ◽

2007 ◽

pp. 83-106

Keyword(s):

Algebraic Multiplicity

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An approximate version of Jackson’s conjecture

Combinatorics Probability Computing ◽

10.1017/s0963548320000152 ◽

2020 ◽

Vol 29 (6) ◽

pp. 886-899

Author(s):

Anita Liebenau ◽

Yanitsa Pehova

Keyword(s):

Bipartite Graph ◽

Small Proportion ◽

Complete Bipartite Graph ◽

Hamilton Cycle ◽

Hamilton Cycles ◽

Edge Disjoint ◽

Bipartite Digraph ◽

Bipartite Tournament ◽

Approximate Version

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

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Finding a Longest Alternating Cycle in a 2-edge-coloured Complete Graph is in RP

Combinatorics Probability Computing ◽

10.1017/s0963548300002054 ◽

1996 ◽

Vol 5 (3) ◽

pp. 297-306 ◽

Cited By ~ 10

Author(s):

Rachid Saad

Keyword(s):

Complete Graph ◽

Strong Evidence ◽

General Setting ◽

Maximum Length ◽

Sufficient Condition ◽

Complete Graphs ◽

Random Polynomial ◽

Exact Matching ◽

Matching Problem ◽

Bipartite Tournament

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

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