On maximum matchings in 5-regular and 6-regular multigraphs. (English) Zbl 1504.05226
Summary: V. V. Mkrtchyan et al. [ibid. 310, No. 10–11, 1588–1613 (2010; Zbl 1200.05178)] made the conjecture that every graph \(G\) with \({\Delta} (G) - \delta (G) \leq 1\) has a maximum matching \(M\) such that no two vertices uncovered by \(M\) share a neighbor. The results obtained by Mkrtchyan et al. [loc.cit.], C. Picouleau [ibid. 310, No. 24, 3646–3647 (2010; Zbl 1200.05181)], P. A. Petrosyan [ibid. 318, 58–61 (2014; Zbl 1281.05112)] and D. Ye [ibid. 341, No. 5, 1195–1198 (2018; Zbl 1383.05265)] leave the conjecture unknown only for 5-regular and 6-regular multigraphs. In this paper, we confirm the conjecture for these two cases.
MSC:
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
| 05C35 | Extremal problems in graph theory |
References:
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