For a 2-connected cubic graph G, the perfect matching polytope P(G) of G contains a special point [...] xc=(13,13,…,13) . The core index ϕ(P(G)) of the polytope P(G) is the minimum number of vertices of P(G) whose convex hull contains xc. The Fulkerson’s conjecture asserts that every 2-connected cubic graph G has six perfect matchings such that each edge appears in exactly two of them, namely, there are six vertices of P(G) such that xc is the convex combination of them, which implies that ϕ(P(G)) ≤ 6. It turns out that the latter assertion in turn implies the Fan-Raspaud conjecture: In every 2-connected cubic graph G, there are three perfect matchings M1, M2, and M3 such that M1 ∩ M2 ∩ M3 = ∅. In this paper we prove the Fan-Raspaud conjecture for ϕ(P(G)) ≤ 12 with certain dimensional conditions.
@article{bwmeta1.element.doi-10_7151_dmgt_2001,
author = {Xiumei Wang and Yixun Lin},
title = {Core Index of Perfect Matching Polytope for a 2-Connected Cubic Graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {38},
year = {2018},
pages = {189-201},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_2001}
}
Xiumei Wang; Yixun Lin. Core Index of Perfect Matching Polytope for a 2-Connected Cubic Graph. Discussiones Mathematicae Graph Theory, Tome 38 (2018) pp. 189-201. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_2001/