Tutte's 5-Flow Conjecture for Highly Cyclically Connected Cubic Graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $\omega$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic graphs with $\omega \geq 2$. We show that if a cubic graph $G$ has no edge cut with fewer than ${5/2} \omega - 1$ edges that separates two odd cycles of a minimum 2-factor of $G$, then $G$ has a nowhere-zero 5-flow. This implies that if a cubic graph $G$ is cyclically $n$-edge connected and $n \geq {5/2} \omega - 1$, then $G$ has a nowhere-zero 5-flow

A note on nowhere-zero 3-flow and Z_3-connectivity

There are many major open problems in integer flow theory, such as Tutte's 3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero 3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is $Z_3$-connected and Kochol's conjecture that every bridgeless graph with at most three 3-edge-cuts admits a nowhere-zero 3-flow (an equivalent version of 3-flow conjecture). Thomassen proved that every 8-edge-connected graph is $Z_3$-connected and therefore admits a nowhere-zero 3-flow. Furthermore, Lov$\acute{a}$sz, Thomassen, Wu and Zhang improved Thomassen's result to 6-edge-connected graphs. In this paper, we prove that: (1) Every 4-edge-connected graph with at most seven 5-edge-cuts admits a nowhere-zero 3-flow. (2) Every bridgeless graph containing no 5-edge-cuts but at most three 3-edge-cuts admits a nowhere-zero 3-flow. (3) Every 5-edge-connected graph with at most five 5-edge-cuts is $Z_3$-connected. Our main theorems are partial results to Tutte's 3-flow conjecture, Kochol's conjecture and Jaeger et al.'s conjecture, respectively.Comment: 10 pages. Typos correcte

Nowhere-Zero Flow Polynomials

In this article we introduce the flow polynomial of a digraph and use it to study nowhere-zero flows from a commutative algebraic perspective. Using Hilbert's Nullstellensatz, we establish a relation between nowhere-zero flows and dual flows. For planar graphs this gives a relation between nowhere-zero flows and flows of their planar duals. It also yields an appealing proof that every bridgeless triangulated graph has a nowhere-zero four-flow

Nowhere-zero 8-flows in cyclically 5-edge-connected, flow-admissible signed graphs

In 1983, Bouchet proved that every bidirected graph with a nowhere-zero integer-flow has a nowhere-zero 216-flow, and conjectured that 216 could be replaced with 6. This paper shows that for cyclically 5-edge-connected bidirected graphs that number can be replaced with 8.Comment: 14 page

Nowhere-Zero 5-Flows On Cubic Graphs with Oddness 4

Tutte’s 5-flow conjecture from 1954 states that every bridge- less graph has a nowhere-zero 5-flow. It suffices to prove the conjecture for cyclically 6-edge-connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow. This implies that every minimum counterexample to the 5-flow conjecture has oddness at least 6

Flows on Signed Graphs

This dissertation focuses on integer flow problems within specific signed graphs. The theory of integer flows, which serves as a dual problem to vertex coloring of planar graphs, was initially introduced by Tutte as a tool related to the Four-Color Theorem. This theory has been extended to signed graphs. In 1983, Bouchet proposed a conjecture asserting that every flow-admissible signed graph admits a nowhere-zero 6-flow. To narrow dawn the focus, we investigate cubic signed graphs in Chapter 2. We prove that every flow-admissible 3-edge-colorable cubic signed graph admits a nowhere-zero 10-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero 10-flow. As a byproduct of this research, we also demonstrate that every flow-admissible hamiltonian signed graph can admit a nowhere-zero 8-flow. In Chapter 3, we delve into triangularly connected signed graphs. Here, A triangle-path in a graph G is defined as a sequence of distinct triangles $T_1,T_2,\ldots,T_m$ in G such that for any i, j with $1\leq i \u3c j \leq m$, $|E(T_i)\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j \u3e i+1$. We categorize a connected graph $G$ as triangularly connected if it can be demonstrated that for any two nonparallel edges $e$ and $e\u27$, there exists a triangle-path $T_1T_2\cdots T_m$ such that $e\in E(T_1)$ and $e\u27\in E(T_m)$. For ordinary graphs, Fan {\it et al.} characterized all triangularly connected graphs that admit nowhere-zero $3$-flows or $4$-flows. Corollaries of this result extended to integer flow in certain families of ordinary graphs, such as locally connected graphs due to Lai and certain types of products of graphs due to Imrich et al. In this dissertation, we extend Fan\u27s result for triangularly connected graphs to signed graphs. We proved that a flow-admissible triangularly connected signed graph $(G,\sigma)$ admits a nowhere-zero $4$-flow if and only if $(G,\sigma)$ is not the wheel $W_5$ associated with a specific signature. Moreover, this result is proven to be sharp since we identify infinitely many unbalanced triangularly connected signed graphs that can admit a nowhere-zero 4-flow but not 3-flow.\\ Chapter 4 investigates integer flow problems within $K_4$-minor free signed graphs. A minor of a graph $G$ refers to any graph that can be derived from $G$ through a series of vertex and edge deletions and edge contractions. A graph is considered $K_4$-minor free if $K_4$ is not a minor of $G$. While Bouchet\u27s conjecture is known to be tight for some signed graphs with a flow number of 6. Kompi\v{s}ov\\u27{a} and M\\u27{a}\v{c}ajov\\u27{a} extended those signed graph with a specific signature to a family \M, and they also put forward a conjecture that suggests if a flow-admissible signed graph does not admit a nowhere-zero 5-flow, then it belongs to \M. In this dissertation, we delve into the members in \M that are $K_4$-minor free, designating this subfamily as $\N$. We provide a proof demonstrating that every flow-admissible, $K_4$-minor free signed graph admits a nowhere-zero 5-flow if and only if it does not belong to the specified family $\N$

Flows on Simplicial Complexes

Given a graph $G$, the number of nowhere-zero \ZZ_q-flows $\phi_G(q)$ is known to be a polynomial in $q$. We extend the definition of nowhere-zero \ZZ_q-flows to simplicial complexes $\Delta$ of dimension greater than one, and prove the polynomiality of the corresponding function $\phi_{\Delta}(q)$ for certain $q$ and certain subclasses of simplicial complexes.Comment: 10 pages, to appear in Discrete Mathematics and Theoretical Computer Science (proceedings of FPSAC'12