Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A Proof of the Two-path Conjecture. | A Proof of the Two-path Conjecture Herbert Fleischner Institute of Discrete Mathematics Austrian Academy of Sciences Sonnenfelsgasse 19 A-1010 Vienna Austria EU Robert R. Molina Department of Mathematics and Computer Science Alma College 614 W. Superior St. Alma MI 48801 molina@ Ken W. Smith Department of Mathematics Central Michigan University Mt. Pleasant MI 48859 Douglas B. West Department of Mathematics University of Illinois 1409 W. Green St. Urbana IL 61801-2975 west@ AMS Subject Classification 05C38 Submitted January 24 2002 Accepted March 13 2002 Abstract Let G be a connected graph that is the edge-disjoint union of two paths of length n where n 2. Using a result of Thomason on decompositions of 4-regular graphs into pairs of Hamiltonian cycles we prove that G has a third path of length n. THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 N4 1 The two-path conjecture states that if a graph G is the edge-disjoint union of two paths of length n with at least one common vertex then the graph has a third subgraph that is also a path of length n. For example the complete graph K4 is an edge-disjoint union of two paths of length 3 each path meeting the other in four vertices. The cycle C6 is the edge-disjoint union of two paths of length 3 with common endpoints. In the first case the graph has twelve paths of length 3 in the second there are six such paths. The two-path conjecture arose in a problem on randomly decomposable graphs. An H-decomposition of a graph G is a family of edge disjoint H-subgraphs of G whose union is G. An H-decomposable graph G is randomly H-decomposable if any edge disjoint family of H-subgraphs of G can be extended to an H-decomposition of G. This concept was introduced by Ruiz in 7 . Randomly Pn-decomposable graphs were studied in 1 5 6 4 . In attempting to classify randomly Pn-decomposable graphs in 5 and 6 it was necessary to know whether the edge-disjoint union of
