Geodesic
Core Index of Perfect Matching Polytope for a 2-Connected Cubic Graph
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 189-201

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For a 2-connected cubic graph G, the perfect matching polytope P(G) of G contains a special point x^c = ( 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 x^c. 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 x^c 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 M_1, M_2, and M_3 such that M_1 ∩ M_2 ∩ M_3 = ∅. In this paper we prove the Fan-Raspaud conjecture for ϕ(P(G)) ≤ 12 with certain dimensional conditions.
Keywords: Fulkerson’s conjecture, Fan-Raspaud conjecture, cubic graph, perfect matching polytope, core index 
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Wang, Xiumei; Lin, Yixun. Core Index of Perfect Matching Polytope for a 2-Connected Cubic Graph. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 189-201. http://geodesic.mathdoc.fr/item/DMGT_2018_38_1_a15/