Geodesic
Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets
Michael Anastos ; David Fabian ; Alp Müyesser1 ; Tibor Szabó
1University College London
The electronic journal of combinatorics, Tome 30 (2023) no. 3
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We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with $|M\cap M_i|=a_i$ for each $i\in \{1,2,3\}$. The bound $n-2$ in the theorem is best possible in general.We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.
DOI : 10.37236/11714
Classification : 05C70, 05C35, 05B15
Mots-clés : Ryser-Brualdi-Stein conjecture

Michael Anastos    ; David Fabian    ; Alp Müyesser  1   ; Tibor Szabó 

1 University College London 
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     author = {Michael Anastos and David Fabian and Alp M\"uyesser and Tibor Szab\'o},
     title = {Splitting matchings and the {Ryser-Brualdi-Stein} conjecture for multisets},
     journal = {The electronic journal of combinatorics},
     year = {2023},
     volume = {30},
     number = {3},
     doi = {10.37236/11714},
     zbl = {1519.05198},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/11714/}
}
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Michael Anastos; David Fabian; Alp Müyesser; Tibor Szabó. Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets. The electronic journal of combinatorics, Tome 30 (2023) no. 3. doi: 10.37236/11714

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