Perfect 1-factorizations

Abstract

Let G be a graph with vertex-set V = V(G) and edge-set E = E(G). A 1-factor of G (also called perfect matching) is a factor of G of degree 1, that is, a set of pairwise disjoint edges which partitions V. A 1-factorization of G is a partition of its edge-set E into 1-factors. For a graph G to have a 1-factor, |V(G)| must be even, and for a graph G to admit a 1-factorization, G must be regular of degree r, 1 ≤ r ≤ |V| − 1. One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs. A 1-factorization of G is said to be perfect if the union of any two of its distinct 1-factors is a Hamiltonian cycle of G. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs. It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

There are 3155 nonisomorphic perfect one‐factorizations of K16

“…The exact number, P n (2 ), of all pairwise nonisomorphic perfect one-factorizations of the complete graph K n 2 has been known only for ≤ n P P P P 2 14: (4) = (6) = (8) = (10) = 1 (cf. [20,22]), P (12) = 5 [16], and P (14) = 23 [6]. In this paper, we establish that P (16) = 3155.…”

“…All other known examples of perfect one-factorizations of K n 2 have been constructed separately using various methods; cf. [20]. The perfect one-factorization conjecture, posed by Kotzig [11], claims the existence of perfect one-factorizations for every even order ≥ n 2 4 of the complete graph K n 2 ; it still seems to be far from proven.…”

“…In particular, the smallest order for which the conjecture remains unsettled is 64 [17]. For surveys on perfect one-factorizations, see [20,22].…”

Perfect 1-Factorisations Of

“…Throughout, we will use the abbreviation P1F for perfect 1-factorisation. For background reading on P1Fs, including definitions of terms not included here, we refer to [8][9][10].…”

A perfect one‐factorisation of

“…By April 2007, Ian Wanless had reported the following additional perfect one-factorisations at [17]: 1295030, 2248092, 2476100, 2685620, 3307950, 3442952, 4657464, 5735340, 6436344, 1030302, 2048384, 4330748, 6967872, 7880600, 9393932, 11089568, 11697084, 13651920, 15813252, 18191448, 19902512, 22665188. Eric Seah published a survey article about perfect 1-factorisations and their properties in 1991 [12], and they are also mentioned by Walter Wallis in chapters of two books printed in 1992 and 1997 (see Section 8 of [13] and Chapter 16 of [14]). A more recent survey regarding perfect 1-factorisations, written by Alex Rosa, is forthcoming and should be consulted for further details about their history and theoretical advances [10]. In the eleven years that have passed since the orders listed in the previous paragraph were published in [2,17], only one new value has been confirmed, namely 52, which was established by Adam Wolfe ten years ago [19].…”