1,308 research outputs found
The topology of competitively constructed graphs
We consider a simple game, the -regular graph game, in which players take
turns adding edges to an initially empty graph subject to the constraint that
the degrees of vertices cannot exceed . We show a sharp topological
threshold for this game: for the case a player can ensure the resulting
graph is planar, while for the case , a player can force the appearance of
arbitrarily large clique minors.Comment: 9 pages, 2 figure
On Hardness of the Joint Crossing Number
The Joint Crossing Number problem asks for a simultaneous embedding of two
disjoint graphs into one surface such that the number of edge crossings
(between the two graphs) is minimized. It was introduced by Negami in 2001 in
connection with diagonal flips in triangulations of surfaces, and subsequently
investigated in a general form for small-genus surfaces. We prove that all of
the commonly considered variants of this problem are NP-hard already in the
orientable surface of genus 6, by a reduction from a special variant of the
anchored crossing number problem of Cabello and Mohar
Handling Handles: Nonplanar Integrability in Supersymmetric Yang-Mills Theory
We propose an integrability setup for the computation of correlation
functions of gauge-invariant operators in supersymmetric
Yang-Mills theory at higher orders in the large genus expansion
and at any order in the 't Hooft coupling . In
this multi-step proposal, one polygonizes the string worldsheet in all possible
ways, hexagonalizes all resulting polygons, and sprinkles mirror particles over
all hexagon junctions to obtain the full correlator. We test our
integrability-based conjecture against a non-planar four-point correlator of
large half-BPS operators at one and two loops.Comment: 6 pages, 4 figures; v2: updated references, typos, minor improvements
(published version
On building 4-critical plane and projective plane multiwheels from odd wheels
We build unbounded classes of plane and projective plane multiwheels that are
4-critical that are received summing odd wheels as edge sums modulo two. These
classes can be considered as ascending from single common graph that can be
received as edge sum modulo two of the octahedron graph O and the minimal wheel
W3. All graphs of these classes belong to 2n-2-edges-class of graphs, among
which are those that quadrangulate projective plane, i.e., graphs from
Gr\"otzsch class, received applying Mycielski's Construction to odd cycle.Comment: 10 page
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