On the maximum degree of path-pairable planar graphs
Ant\'onio Gir\~ao, G\'abor M\'esz\'aros, Kamil Popielarz, Richard, Snyder
TL;DR
This paper proves that in any path-pairable planar graph with n vertices, there must be a vertex with degree proportional to n, highlighting a fundamental property of such graphs.
Contribution
It establishes a lower bound on the maximum degree in path-pairable planar graphs, revealing a key structural limitation.
Findings
Any n-vertex path-pairable planar graph has a vertex with degree linear in n.
The result constrains the possible degree distributions in such graphs.
Abstract
A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any -vertex path-pairable planar graph must contain a vertex of degree linear in .
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On the maximum degree of path-pairable planar graphs
António Girão Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK; [email protected]
Gábor Mészáros Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee; [email protected]
Kamil Popielarz Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee; [email protected]
Richard Snyder Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee; [email protected]
Abstract
A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any -vertex path-pairable planar graph must contain a vertex of degree linear in .