Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Triangulations and the Haj´s Conjecture o. | Triangulations and the Hajos Conjecture Bojan Mohar Department of Mathematics University of Ljubljana 1000 Ljubljana Slovenia Submitted Apr 18 2005 Accepted Sep 1 2005 Published Sep 14 2005 Mathematics Subject Classifications 05C10 05C15 Abstract The Hajos Conjecture was disproved in 1979 by Catlin. Recently Thomassen showed that there are many ways that Hajos conjecture can go wrong. On the other hand he observed that locally planar graphs and triangulations of the projective plane and the torus satisfy Hajos Conjecture and he conjectured that the same holds for arbitrary triangulations of closed surfaces. In this note we disprove the conjecture and show that there are different reasons why the Hajos Conjecture fails also for triangulations. 1 Introduction Hajos conjecture claims that every graph whose chromatic number is at least k contains a subdivision of Kk the complete graph of order k. The conjecture has been proved for k 4 by Dirac 3 while for k 5 it yields a strengthening of the Four Color Theorem which is still open. The conjecture was disproved for all k 7 by Catlin 2 . Soon after that Erdos and Fajtlowicz 4 proved that the conjecture is false for almost all graphs. Recently Thomassen 11 revived the interest in Hajos conjecture by showing that there is a great variety of reasons why Hajos conjecture can be wrong. At the end of this interesting work Thomassen observed that the Hajos conjecture could be true to some limited extent. Maybe it holds in the setting under whose influence it was originally formulated related to the Four Color Conjecture . For instance it holds for graphs embedded in any fixed surface with sufficiently large edge-width . when all noncontractible cycles are long . Therefore it seems plausible to propose Supported in part by the Ministry of Education Science and Sport of Slovenia Research Project J1-0502-0101 and Research Program P1-0297. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N15 1 Conjecture
