The Parity of a Thicket
A thicket in a graph $G$ is defined as a set of even circuits such that every edge lies in an even number of them. If $G$ is directed, then each circuit in the thicket has a well defined directed parity. The parity of the thicket is the sum of the parities of its members, and is independent of the orientation of $G$. We study the problem of determining the parity of a thicket $\mathcal{T}$ in terms of structural properties of $\mathcal{T}$. Specifically, we reduce the problem to studying the case where the underlying graph $G$ is cubic. In this case we solve the problem if $|\mathcal{T}| = 3$ or $G$ is bipartite. Some applications to the problem of characterising Pfaffian graphs are also considered.
1988 ◽
Vol 447
(3)
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pp. 103-116
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1984 ◽
Vol 127
(1-3)
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pp. 214-218
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1998 ◽
Vol 08
(PR2)
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pp. Pr2-47-Pr2-50
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2000 ◽
Vol 10
(PR7)
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pp. Pr7-95-Pr7-98
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2013 ◽
Vol 51
(9)
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pp. 691-699
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1962 ◽
Vol 78
(12)
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pp. 579-617
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2007 ◽
Vol 2007
(suppl_26)
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pp. 503-508