@article{59169,
  abstract     = {{An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r-graphs. An H-coloring of G is a mapping such that each r adjacent edges of G are mapped to r adjacent edges of H. For every , let be an inclusion-wise minimal set of connected r-graphs, such that for every connected r-graph G there is an which colors G. The Petersen Coloring Conjecture states that consists of the Petersen graph P. We show that if true, then this is a very exclusive situation. Our main result is that either or is an infinite set and if , then is an infinite set. In particular, for all , is unique. We first characterize and then prove that if contains more than one element, then it is an infinite set. To obtain our main result we show that contains the smallest r-graphs of class 2 and the smallest poorly matchable r-graphs, and we determine the smallest r-graphs of class 2.}},
  author       = {{Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak H.}},
  issn         = {{0209-9683}},
  journal      = {{Combinatorica}},
  number       = {{2}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Sets of r-Graphs that Color All r-Graphs}}},
  doi          = {{10.1007/s00493-025-00144-4}},
  volume       = {{45}},
  year         = {{2025}},
}

@article{49905,
  abstract     = {{For 0 ≤ t ≤ r let m(t, r) be the maximum number s such that every t-edge-connected r-graph has s pairwise disjoint perfect matchings. There are only a few values of m(t, r) known, for instance m(3, 3) = m(4, r) = 1, and m(t, r) ≤ r − 2 for all t  = 5,
and m(t, r) ≤ r − 3 if r is even. We prove that m(2l, r) ≤ 3l − 6 for every l ≥ 3 and r ≥ 2l.}},
  author       = {{Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}},
  issn         = {{0209-9683}},
  journal      = {{Combinatorica}},
  keywords     = {{Computational Mathematics, Discrete Mathematics and Combinatorics}},
  pages        = {{429--440}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs}}},
  doi          = {{10.1007/s00493-023-00078-9}},
  volume       = {{44}},
  year         = {{2024}},
}