Accola theorem on hyperelliptic graphs

Harmonic Morphisms and Hyperelliptic Graphs

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical RiemannHurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also disc...

Ramsey’s Theorem on Graphs

Imagine that you have 6 people at a party. We assume that, for every pair of them, either THEY KNOW EACH OTHER or NEITHER OF THEM KNOWS THE OTHER. So we are assuming that if x knows y, then y knows x. Claim: Either there are at least 3 people all of whom know one another, or there are at least 3 people no two of whom know each other (or both). Proof of Claim: Let the people be p1, p2, p3, p4, p...

A Theorem on Planar Graphs

Recognizing hyperelliptic graphs in polynomial time

Recently, a new set of multigraph parameters was defined, called “gonalities”. Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1 are trees. We consider so-called “hyperelliptic graphs” (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigr...

An Explicit Theorem of the Square for Hyperelliptic Jacobians

LetA be an abelian variety over a field k, D a symmetric divisor onA, s and d the sum and difference maps fromA×A intoA, andp1 andp2 the projections onto the first and second factors. The theorem of the square and the seesaw principle [M1, Secs. 5, 6] guarantee that there exists a function f(u, v) on A×A (determined up to constant multiples) with divisor s∗D+ d ∗D− 2p∗ 1D− 2p∗2D. Since this fun...