Properly Coloured Hamiltonian Paths in Edge-coloured Complete Graphs
نویسندگان
چکیده
We consider edge-coloured complete graphs. A path or cycle Q is called properly coloured (PC) if any two adjacent edges of Q differ in colour. Our note is inspired by the following conjecture by B. Bollobás and P. Erdős (1976) : if G is an edge-coloured complete graph on n vertices in which the maximum monochromatic degree of every vertex is less than bn/2c, then G contains a PC Hamiltonian cycle. We prove that if an edge-coloured complete graph contains a PC 2-factor then it has a PC Hamiltonian path. R. Häggkvist (1996) announced that every edge-coloured complete graph satisfying Bollobás-Erdős condition contains a PC 2-factor. These two results imply that every edge-coloured complete graph satisfying Bollobás-Erdős condition has a PC Hamiltonian path.
منابع مشابه
Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost n. In this paper, improving on several e...
متن کاملGadget Graphs for Edge-coloured Graphs
In this paper, we consider properly edge-coloured (PC) paths and cycles in edge-coloured graphs. We consider a family of transformations of an edge-coloured graph G into an ordinary graph that allow us to check the existence PC cycles and PC (s, t)-paths in G and, if they exist, to find shortest ones among them. We raise a problem of finding the optimal transformation and consider a possible so...
متن کاملRainbow spanning trees in properly coloured complete graphs
In this short note, we study pairwise edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, where a graph is rainbow if its edges have distinct colours. Brualdi and Hollingsworth conjectured that every Kn properly edge-coloured by n−1 colours has n/2 edge-disjoint rainbow spanning trees. Kaneko, Kano and Suzuki later suggested this should hold for every properly edge-c...
متن کاملProper Hamiltonian Paths in Edge-Coloured Multigraphs
Given a c-edge-coloured multigraph, a proper Hamiltonian path is a path that contains all the vertices of the multigraph such that no two adjacent edges have the same colour. In this work we establish sufficient conditions for an edge-coloured multigraph to guarantee the existence of a proper Hamiltonian path, involving various parameters as the number of edges, the number of colours, the rainb...
متن کاملLarge Rainbow Matchings in Edge-Coloured Graphs
A rainbow subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k 4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least k/2 . A properly edge-coloured K4 has no such matching, which motivates ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Applied Mathematics
دوره 82 شماره
صفحات -
تاریخ انتشار 1998