34,293 research outputs found
Facial non-repetitive edge-colouring of plane graphs
A sequence such that for all , is called a {\em repetition}. A sequence is called {\em non-repetitive} if no {\it block} (i.e. subsequence of consecutive terms of ) is a repetition. Let be a graph whose edges are coloured. A trail is called {\em non-repetitive} if the sequence of colours of its edges is non-repetitive. If is a plane graph, a {\em facial non-repetitive edge-colouring} of is an edge-colouring such that any {\it facial trail} (i.e. trail of consecutive edges on the boundary walk of a face) is non-repetitive. We denote the minimum number of colours of a facial non-repetitive edge-colouring of . In this paper, we show that for any plane graph . We also get better upper bounds for in the cases when is a tree, a plane triangulation, a simple -connected plane graph, a hamiltonian plane graph, an outerplanar graph or a Halin graph. The bound for trees is tight
Types of triangle in plane Hamiltonian triangulations and applications to domination and k-walks
We investigate the minimum number t(0)(G) of faces in a Hamiltonian triangulation G so that any Hamiltonian cycle C of G has at least t(0)(G) faces that do not contain an edge of C. We prove upper and lower bounds on the maximum of these numbers for all triangulations with a fixed number of facial triangles. Such triangles play an important role when Hamiltonian cycles in triangulations with 3-cuts are constructed from smaller Hamiltonian cycles of 4-connected subgraphs. We also present results linking the number of these triangles to the length of 3-walks in a class of triangulation and to the domination number
Polyhedra with few 3-cuts are hamiltonian
In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In
this article, we will generalize this result and prove that polyhedra with at
most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this
result for the subclass of triangulations. We also prove that polyhedra with at
most four 3-cuts have a hamiltonian path. It is well known that for each non-hamiltonian polyhedra with 3-cuts exist. We give computational
results on lower bounds on the order of a possible non-hamiltonian polyhedron
for the remaining open cases of polyhedra with four or five 3-cuts.Comment: 21 pages; changed titl
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