One‐factorizations of the complete graph—A survey

Abstract

We survey known results on the existence and enumeration of many kinds of l-factorizations of the complete graph. We also mention briefly some related questions and topics. as well as applications.

Indecomposable 1-factorizations of the complete multigraph

“…A comprehensive survey of research on 1-factorizations of complete graphs is given in [3]. If no two members of F are identical as 1-factors (i.e., no 1-factors are "repeated"), the l-factorization is said to be simple.…”

“…Consider all of the pairs {x, y), x, y e Z 2n _ x having \x -y\ = 1. There are 2« -1 [3] such pairs, each of which is contained in exactly X factors of F'. On the other hand, F' contains m 1-factors for some m < 2n -1 whose edges {x, y) are such that \x -y\ = 1, and each of these 1-factors contains 2n -2 such edges.…”

“…Simple IOF(1Q, 3) {0,oo} {1,8} {2,3} {4,5} {6,7} {0,<»} {1,3} {2,7} {4,6} {5,8} {0,oo} {1,6} {2,5} {3,8} {4,7} Simple 7OF(12,3) {0,oo} {1,7} {2,3} {4,5} {6,10} {8,9} {0,oo} {1,7} {2,9} {3,5} {4,6} {8,10} {0,oo} {1,4} {2,7} {3,10} {5,8} {6,9}…”

On the application of graph colouring techniques in round-robin sports scheduling

“…Note that many previous methods for sports timetabling have also used graphs for representing round-robins, though these have usually been of a different form [11,31,39]. Specifically, in such cases a complete graph K n is used, with each vertex denoting a team and each edge denoting a match between its two endpoints.…”

Scheduling A Major College Basketball Conference

“…DeWerra (1980,1988) and others in combinatorial design (Mendelsohn and Rosa 1985) have made great advances in determining feasible timetables, but there is no known characterization of all feasible timetables. The known timetables all revolve around some particular property, like maximizing or minimizing the number of alternations of home and away in the patterns, which may not be critical or appropriate in any particular application.…”