#### Semester

Fall

#### Date of Graduation

2020

#### Document Type

Thesis

#### Degree Type

PhD

#### College

Eberly College of Arts and Sciences

#### Department

Mathematics

#### Committee Chair

Hong-Jian Lai

#### Committee Member

John Goldwasser

#### Committee Member

K. Subramani

#### Committee Member

Rong Luo

#### Committee Member

Jerzy Wojciechowski

#### Abstract

This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte's 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte's 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. Extending Tutte's flows conjectures, Jaeger's Circular Flow Conjecture (1981) says every 4k-edge-connected graph admits a modulo (2k+1)-orientation, that is, an orientation such that the indegree is congruent to outdegree modulo (2k+1) at every vertex. Note that the k=1 case of Circular Flow Conjecture coincides with the 3-Flow Conjecture, and the case of k=2 implies the 5-Flow Conjecture. This work is devoted to providing some partial results on these problems.

In Chapter 2, we study the problem of modulo 5-orientation for given multigraphic degree sequences. We prove that a multigraphic degree sequence d=(d_{1},..., d_{n}) has a realization G with a modulo 5-orientation if and only if d_{i}≤1,3 for each i. In addition, we show that every multigraphic sequence d=(d_{1},..., d_{n}) with min_{{1≤i≤n}}d_{i}≥9 has a 9-edge-connected realization that admits a modulo 5-orientation for every possible boundary function. Jaeger conjectured that every 9-edge-connected multigraph admits a modulo 5-orientation, whose truth would imply Tutte's 5-Flow Conjecture. Consequently, this supports the conjecture of Jaeger.

In Chapter 3, we show that there are essentially finite many exceptions for graphs with bounded matching numbers not admitting any modulo (2k+1)-orientations for any positive integers t. We additionally characterize all infinite many graphs with bounded matching numbers but without a nowhere-zero 3-flow. This partially supports Jaeger's Circular Flow Conjecture and Tutte's 3-Flow Conjecture.

In 2018, Esperet, De Verclos, Le and Thomass introduced the problem of weighted modulo orientations of graphs and indicated that this problem is closely related to modulo orientations of graphs, including Tutte's 3-Flow Conjecture. In Chapter 4 and Chapter 5, utilizing properties of additive bases and contractible configurations, we reduced the Esperet et al's edge-connectivity lower bound for some (signed) graphs families including planar graphs, complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.

In Chapter 6, we show that the assertion of Jaeger's Circular Flow Conjecture with k=2 holds asymptotically almost surely for random 9-regular graphs.

#### Recommended Citation

Liu, Jianbing, "Weighted Modulo Orientations of Graphs" (2020). *Graduate Theses, Dissertations, and Problem Reports*. 7757.

https://researchrepository.wvu.edu/etd/7757