#### Semester

Summer

#### Date of Graduation

2012

#### Document Type

Dissertation

#### Degree Type

PhD

#### College

Eberly College of Arts and Sciences

#### Department

Mathematics

#### Committee Chair

Cun-Quan Zhang.

#### Abstract

Tutte's 3-flow conjecture (1970's) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. A graph G admits a nowhere-zero 3-flow if and only if G has an orientation such that the out-degree equals the in-degree modulo 3 for every vertex. In the 1980ies Jaeger suggested some related conjectures. The generalized conjecture to modulo k-orientations, called circular flow conjecture, says that, for every odd natural number k, every (2k-2)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex. And the weaker conjecture he made, known as the weak 3-flow conjecture where he suggests that the constant 4 is replaced by any larger constant.;The weak version of the circular flow conjecture and the weak 3-flow conjecture are verified by Thomassen (JCTB 2012) recently. He proved that, for every odd natural number k, every (2k 2 + k)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex and for k = 3 the edge-connectivity 8 suffices. Those proofs are refined in this paper to give the same conclusions for 9 k-edge-connected graphs and for 6-edge-connected graphs when k = 3 respectively.

#### Recommended Citation

Wu, Yezhou, "Integer flows and Modulo Orientations" (2012). *Graduate Theses, Dissertations, and Problem Reports*. 3566.

https://researchrepository.wvu.edu/etd/3566