Properly colored Hamilton cycles in edge-colored complete graphs

Abstract

ABSTRACT:It is shown that, for ⑀ ) 0 and n ) n ⑀ , any complete graph K on n vertices 0

Characterization of edge‐colored complete graphs with properly colored Hamilton paths

“…which are properly colored and satisfy (1). Since b(C 1 , P 1 ) < b(C, P ) holds, our minimality assumption yields a PCHP as in Claim A, which is a contradiction.…”

“…The best result so far for 'small' values of n is by J. Shearer [12]: 7∆ mon < n/2 guarantees the existence of a properly colored Hamilton cycle. The best result so far for large values of n is due to N. Alon and G. Gutin [1]: For every > 0 and n = n large enough, ∆ mon ≤ (1 − 1 √ 2 − ) n/2 implies the existence of a properly colored Hamilton cycle.…”

Cycles and paths in edge‐colored graphs with given degrees

“…Bollobás and Erdös in [8], conjectured that if the monochromatic degree of every vertex in K c n is strictly less than n 2 , then K c n contains a properly edge-colored Hamiltonian cycle, for any c ≥ 3. This conjecture was partially proved in [1] by using an advanced probabilistic method. The conjecture below is a weaker version of that conjecture by Bollobás and Erdös for regular edge-colored complete graphs, and perhaps an easier one to prove.…”

Properly Coloured Cycles and Paths: Results and Open Problems

“…Improving some previous results on this conjecture, Shearer [18] showed that if 7∆ mon (K c n ) < n, then K c n has a PC Hamilton cycle. So far, the best asymptotic estimate was obtained by Alon and Gutin [3]. …”