Abstract
Suppose (Formula presented.), (Formula presented.). A set (Formula presented.) of (Formula presented.) line segments is a matching of (Formula presented.) if (Formula presented.) is the set of endpoints of (Formula presented.). The length of (Formula presented.) is the sum of lengths of (Formula presented.). (Formula presented.) is called a crossing matching if every two segments (Formula presented.) cross. It is shown that a crossing matching (Formula presented.) (if it exists) is longer than any other matching of (Formula presented.). In fact, if (Formula presented.) is a crossing matching of (Formula presented.), and (Formula presented.) is an arbitrary matching of (Formula presented.), then, within the union (Formula presented.), we can find (Formula presented.) polygonal paths that connect (Formula presented.) to (Formula presented.) ((Formula presented.)) and are pairwise almost disjoint. In the same vein: suppose (Formula presented.). A Hamiltonian (Formula presented.) -circuit (Formula presented.) on (Formula presented.) is an asterisk if every two nonadjacent edges of (Formula presented.) cross. It is shown that if (Formula presented.) admits an asterisk (Formula presented.), then (Formula presented.) is longer than any other (Formula presented.) -regular geometric graph on (Formula presented.). This proves an old conjecture by the first author.
| Original language | English |
|---|---|
| Pages (from-to) | 159-169 |
| Number of pages | 11 |
| Journal | Journal of Graph Theory |
| Volume | 94 |
| Issue number | 1 |
| DOIs | |
| State | Published - 1 May 2020 |
Bibliographical note
Publisher Copyright:© 2019 Wiley Periodicals, Inc.
Keywords
- asterisk
- crossing matching
- geometric graphs
- length
- perfect matching