Topic
Halin graph
About: Halin graph is a research topic. Over the lifetime, 193 publications have been published within this topic receiving 1501 citations.
Papers published on a yearly basis
Papers
TL;DR: Even polyhedral decompositions of a trivalent graph have been shown to have a Tait colouring by three colours a, b, c as mentioned in this paper, which is a strong conjecture.
Abstract: A polyhedral decomposition of a finite trivalent graph G is defined as a set of circuits = {C1, C2, … Cm} with the property that every edge of G occurs exactly twice as an edge of some Ck. The decomposition is called even if every Ck is a simple circuit of even length. If G has a Tait colouring by three colours a, b, c then the (a, b), (b, c) and (c, a) circuits obviously form an even polyhedral decomposition. It is shown that the converse is also true: if G has an even polyhedral decomposition then it also has a Tait colouring. This permits an equivalent formulation of the four colour conjecture (and a much stronger conjecture of Branko Grunbaum) in terms of polyhedral decompositions alone.
222 citations
TL;DR: A linear time algorithm for the Steiner problem in Halin networks is presented, providing another example where the recursive structure of the underlying network leads to an efficient algorithm.
42 citations
TL;DR: It is proved that the incidence coloring number for Halin graphs and outerplanar graphs of maximum degree greater than or equal to 4 is equal to the maximum degree of the graph plus 1.
42 citations
TL;DR: It is proved that such a graph on n vertices contains cycles of all lengths l, 3 ≤ ln, except, possibly, for one even value m of l.
Abstract: A Halin graph is a plane graph H = T U C, where T is a plane tree with no vertex of degree two and at least one vertex of degree three or more, and C is a cycle connecting the endvertices of T in the cyclic order determined by the embedding of T. We prove that such a graph on n vertices contains cycles of all lengths l, 3 ≤ ln, except, possibly, for one even value m of l. We prove also that if the tree T contains no vertex of degree three then G is pancyclic.
41 citations
TL;DR: The strong chromatic index of a graph G, denoted by [email protected]^'(G), is the minimum number of colors needed for a strong edge coloring of G.
36 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 9 |
| 2020 | 7 |
| 2019 | 9 |
| 2018 | 4 |
| 2017 | 10 |
| 2016 | 9 |