✦ LIBER ✦

Infinite paths in planar graphs IV, dividing cycles

✍ Scribed by Xingxing Yu

Publisher: John Wiley and Sons
Year: 2006
Tongue: English
Weight: 285 KB
Volume: 53
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.20176

No coin nor oath required. For personal study only.

✦ Synopsis

Nash-Williams conjectured that a 4-connected infinite planar graph contains a spanning 2-way infinite path if, and only if, the deletion of any finite set of vertices results in at most two infinite components. In this article, we prove this conjecture for graphs with no dividing cycles and for graphs with infinitely many vertex disjoint dividing cycles. A cycle in an infinite plane graph is called dividing if both regions of the plane bounded by this cycle contain infinitely many vertices of the graph.

📜 SIMILAR VOLUMES

Spanning paths in infinite planar graphs

✍ Dean, Nathaniel; Thomas, Robin; Yu, Xingxing 📂 Article 📅 1996 🏛 John Wiley and Sons 🌐 English ⚖ 796 KB

Let G be a 4connected infinite planar graph such that the deletion of any finite set of vertices of G results in at most one infinite component. We prove a conjecture of Nash-Williams that G has a 1 -way infinite spanning path. 0 1996 John Wiley & Sons, Inc. [7] has shown that every 4-connected fini

Infinite paths in planar graphs III, 1-w

Infinite paths in planar graphs III, 1-way infinite paths

✍ Xingxing Yu 📂 Article 📅 2006 🏛 John Wiley and Sons 🌐 English ⚖ 336 KB

An infinite graph is 2-indivisible if the deletion of any finite set of vertices from the graph results in exactly one infinite component. Let G be a 4-connected, 2-indivisible, infinite, plane graph. It is known that G contains a spanning 1-way infinite path. In this paper, we prove a stronger resu

Infinite paths in planar graphs I: Graph

Infinite paths in planar graphs I: Graphs with radial nets

✍ Xingxing Yu 📂 Article 📅 2004 🏛 John Wiley and Sons 🌐 English ⚖ 234 KB 👁 1 views

## Abstract Let __G__ be an infinite 4‐connected planar graph such that the deletion of any finite set of vertices from __G__ results in exactly one infinite component. Dean __et al__. proved that either __G__ admits a radial net or a special subgraph of __G__ admits a ladder net, and they used the

Infinite paths in planar graphs II, stru

Infinite paths in planar graphs II, structures and ladder nets

✍ Xingxing Yu 📂 Article 📅 2005 🏛 John Wiley and Sons 🌐 English ⚖ 217 KB 👁 1 views

## Abstract A graph is __k‐indivisible__, where __k__ is a positive integer, if the deletion of any finite set of vertices results in at most __k__ – 1 infinite components. In 1971, Nash‐Williams conjectured that a 4‐connected infinite planar graph contains a spanning 2‐way infinite path if and onl

On paths in planar graphs

✍ Sanders, Daniel P. 📂 Article 📅 1997 🏛 John Wiley and Sons 🌐 English ⚖ 93 KB 👁 2 views

This paper generalizes a theorem of Thomassen on paths in planar graphs. As a corollary, it is shown that every 4-connected planar graph has a Hamilton path between any two specified vertices x, y and containing any specified edge other than xy.

A theorem on paths in planar graphs

✍ Norishige Chiba; Takao Nishizeki 📂 Article 📅 1986 🏛 John Wiley and Sons 🌐 English ⚖ 108 KB 👁 1 views

C. Thomassen extended Tutte's theorem on cycles in planar graphs in the paper "A Theorem on Paths in Planar Graphs". This note corrects a flaw in his proof.