A lower bound for the complex flow number of a graph: A geometric approach

Author:

Mattiolo Davide1,Mazzuoccolo Giuseppe2,Rajník Jozef3,Tabarelli Gloria4

Affiliation:

1. Department of Computer Science KU Leuven Kulak Kortrijk Belgium

2. Dipartimento di Scienze Fisiche, Informatiche e Matematiche Università di Modena e Reggio Emilia Modena Italy

3. Department of Computer Science Comenius University in Bratislava Bratislava Slovakia

4. Dipartimento di Matematica Università di Trento Trento Italy

Abstract

AbstractLet be a real number. A complex nowhere‐zero ‐flow on a graph is an orientation of together with an assignment such that, for all , the Euclidean norm of the complex number lies in the interval and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph , denoted by , is the minimum of the real numbers such that admits a complex nowhere‐zero ‐flow. The exact computation of seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of is known only for families of graphs where a lower bound can be trivially proved. Here, we use geometric and combinatorial arguments to give a nontrivial lower bound for in terms of the odd‐girth of a cubic graph (i.e., the length of a shortest odd cycle) and we show that this lower bound is tight. This result relies on the exact computation of the complex flow number of the wheel graph . In particular, we show that for every odd , the value of arises from one of three suitable configurations of points in the complex plane according to the congruence of modulo 6.

Funder

Fonds Wetenschappelijk Onderzoek

Publisher

Wiley

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5. A unified approach to construct snarks with circular flow number 5

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