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A Correspondence between Plane Planted Chromatic Trees and Generalised Derangements

A Correspondence between Plane Planted Chromatic Trees and Generalised Derangements A CORRESPONDENCE BETWEEN PLANE PLANTED CHROMATIC TREES AND GENERALISED DERANGEMENTS I. P. GOULDE N AND D. M. JACKSON §1. The correspondence A tree in which adjacent vertices are of different colours is called a chromatic tree. If L is a matrix with element l equal to the number of non-roo t vertices of colour i tj and degree j in a chromatic tree t, then L is called the chromatic partition of t. We demonstrate the existence of a remarkable and tantalising correspondence between two apparently unrelated combinatorial sets, namely plane planted chromatic trees and generalised derangements. The derangement may, of course, be used to construct a graph in which no two vertices of the same colour are joined by an edge, but we cannot guarantee that this graph is in fact a tree. Indeed, we have been unable to give a combinatorial characterisation of this one-to-one correspondence. The correspondence allows us to enumerate K-chromatic trees with given chromatic partition. This is a new result. The case K - 2 has been given previously by Tutte [4]. Throughout this paper we use the following notation: the number of vertices of colour / is n, = £ l ; the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Correspondence between Plane Planted Chromatic Trees and Generalised Derangements

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References (3)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/13.1.28
Publisher site
See Article on Publisher Site

Abstract

A CORRESPONDENCE BETWEEN PLANE PLANTED CHROMATIC TREES AND GENERALISED DERANGEMENTS I. P. GOULDE N AND D. M. JACKSON §1. The correspondence A tree in which adjacent vertices are of different colours is called a chromatic tree. If L is a matrix with element l equal to the number of non-roo t vertices of colour i tj and degree j in a chromatic tree t, then L is called the chromatic partition of t. We demonstrate the existence of a remarkable and tantalising correspondence between two apparently unrelated combinatorial sets, namely plane planted chromatic trees and generalised derangements. The derangement may, of course, be used to construct a graph in which no two vertices of the same colour are joined by an edge, but we cannot guarantee that this graph is in fact a tree. Indeed, we have been unable to give a combinatorial characterisation of this one-to-one correspondence. The correspondence allows us to enumerate K-chromatic trees with given chromatic partition. This is a new result. The case K - 2 has been given previously by Tutte [4]. Throughout this paper we use the following notation: the number of vertices of colour / is n, = £ l ; the

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jan 1, 1981

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