Page 313 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 313
GRAPHS, POLYNOMIALS, SURFACES, AND KNOTS (MS-49)
Eulerian and bipartite partial duals
Metrose Metsidik, metrose@163.com
Xinjiang Normal University, China
Huggett and Moffatt characterized all bipartite partial duals of plane graphs in terms of all-
crossing directions of their medial graphs. Then Metsidik and Jin characterized all Eulerian
partial duals of plane graphs in terms of semi-crossing directions of their medial graphs. Plane
graphs are ribbon graphs with genus 0. In this talk, by introducing the notion of modified
medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s
result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of
any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler
than semi-crossing directions.
From matrix pivots to graphs in surfaces: touring combinatorics as
guided by partial duals
Iain Moffatt, iain.moffatt@rhul.ac.uk
Royal Holloway, University of London, United Kingdom
This talk will be accessible to a general mathematical, non-specialist audience.
The concept of the dual of a graph traces back to the very beginnings of graph theory, and
can even be found in work of Euler. Roughy speaking, the dual of a graph drawn on the plane is
formed by switching its vertices and faces. (For example, the dual of a cube is an octahedron.)
Duals are a foundational and well-known concept in graph theory — most undergraduates will
meet them at some point in their studies.
Despite their long history, it has only become apparent that you don’t have to form the dual
of all of the edges of a graph at once, you can just take the dual with respect to some of its
edges. This results in the idea of a “partial dual” — a concept introduced by S. Chmutov in
2009. It arose from work on the Jones polynomial and knot theory.
With such an advance in understanding of such a fundamental construction, it is perhaps
unsurprising that partial duals swiftly led to advances in topological graph theory. However,
and perhaps more importantly, it turns out that the partial duals reach far beyond graphs in
surfaces, are intimately related to several very different areas of combinatorics. Indeed, the
concept of the “partial dual” has appeared in very highly disguised forms in various places in
the literature over the last 60 years.
In this talk, I’ll survey the various appearances and applications of partial duals in graph
theory, pointing out the places where the concept has been hiding in the literature for all these
years. Along the way we’ll encounter various topics in combinatorics such as pivots of ma-
trices, embedded graphs, circle graphs, the Tutte polynomial, knot theory, pivot minors, chord
diagrams, and matroids. The emphasis will be on how the various topics fit together, and on
what is to be gained by switching between the various perspectives.
311
Eulerian and bipartite partial duals
Metrose Metsidik, metrose@163.com
Xinjiang Normal University, China
Huggett and Moffatt characterized all bipartite partial duals of plane graphs in terms of all-
crossing directions of their medial graphs. Then Metsidik and Jin characterized all Eulerian
partial duals of plane graphs in terms of semi-crossing directions of their medial graphs. Plane
graphs are ribbon graphs with genus 0. In this talk, by introducing the notion of modified
medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s
result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of
any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler
than semi-crossing directions.
From matrix pivots to graphs in surfaces: touring combinatorics as
guided by partial duals
Iain Moffatt, iain.moffatt@rhul.ac.uk
Royal Holloway, University of London, United Kingdom
This talk will be accessible to a general mathematical, non-specialist audience.
The concept of the dual of a graph traces back to the very beginnings of graph theory, and
can even be found in work of Euler. Roughy speaking, the dual of a graph drawn on the plane is
formed by switching its vertices and faces. (For example, the dual of a cube is an octahedron.)
Duals are a foundational and well-known concept in graph theory — most undergraduates will
meet them at some point in their studies.
Despite their long history, it has only become apparent that you don’t have to form the dual
of all of the edges of a graph at once, you can just take the dual with respect to some of its
edges. This results in the idea of a “partial dual” — a concept introduced by S. Chmutov in
2009. It arose from work on the Jones polynomial and knot theory.
With such an advance in understanding of such a fundamental construction, it is perhaps
unsurprising that partial duals swiftly led to advances in topological graph theory. However,
and perhaps more importantly, it turns out that the partial duals reach far beyond graphs in
surfaces, are intimately related to several very different areas of combinatorics. Indeed, the
concept of the “partial dual” has appeared in very highly disguised forms in various places in
the literature over the last 60 years.
In this talk, I’ll survey the various appearances and applications of partial duals in graph
theory, pointing out the places where the concept has been hiding in the literature for all these
years. Along the way we’ll encounter various topics in combinatorics such as pivots of ma-
trices, embedded graphs, circle graphs, the Tutte polynomial, knot theory, pivot minors, chord
diagrams, and matroids. The emphasis will be on how the various topics fit together, and on
what is to be gained by switching between the various perspectives.
311
