Weighted Modulo Orientations of Graphs

This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte\u27s 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte\u27s 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. Extending Tutte\u27s flows conjectures, Jaeger\u27s 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=(d1,..., dn) has a realization G with a modulo 5-orientation if and only if di≤1,3 for each i. In addition, we show that every multigraphic sequence d=(d1,..., dn) with min{1≤i≤n}di≥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\u27s 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\u27s Circular Flow Conjecture and Tutte\u27s 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\u27s 3-Flow Conjecture. In Chapter 4 and Chapter 5, utilizing properties of additive bases and contractible configurations, we reduced the Esperet et al\u27s 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\u27s Circular Flow Conjecture with k=2 holds asymptotically almost surely for random 9-regular graphs

The Number of Nowhere-Zero Flows on Graphs and Signed Graphs

A nowhere-zero $k$-flow on a graph $\Gamma$ is a mapping from the edges of $\Gamma$ to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in any fixed orientation of $\Gamma$, at each node the sum of the labels over the edges pointing towards the node equals the sum over the edges pointing away from the node. We show that the existence of an \emph{integral flow polynomial} that counts nowhere-zero $k$-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the flow counting functions, and reciprocity laws that interpret the evaluations of the flow polynomials at negative integers in terms of the combinatorics of the graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.

Flows on Bidirected Graphs

The study of nowhere-zero flows began with a key observation of Tutte that in planar graphs, nowhere-zero k-flows are dual to k-colourings (in the form of k-tensions). Tutte conjectured that every graph without a cut-edge has a nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero 6-flow. For a graph embedded in an orientable surface of higher genus, flows are not dual to colourings, but to local-tensions. By Seymour's theorem, every graph on an orientable surface without the obvious obstruction has a nowhere-zero 6-local-tension. Bouchet conjectured that the same should hold true on non-orientable surfaces. Equivalently, Bouchet conjectured that every bidirected graph with a nowhere-zero $\mathbb{Z}$-flow has a nowhere-zero 6-flow. Our main result establishes that every such graph has a nowhere-zero 12-flow.Comment: 24 pages, 2 figure

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 $\N$

An 8-flow theorem for signed graphs

We prove that a signed graph admits a nowhere-zero $8$-flow provided that it is flow-admissible and the underlying graph admits a nowhere-zero $4$-flow. When combined with the 4-color theorem, this implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero $8$-flow. Our result improves and generalizes previous results of Li et al. (European J. Combin. 108 (2023), 103627), which state that every flow-admissible signed $3$-edge-colorable cubic graph admits a nowhere-zero $10$-flow, and that every flow-admissible signed hamiltonian graph admits a nowhere-zero $8$-flow.Comment: 12 pages, 2 figure

Cycles through two edges in signed graphs

We give a characterization of when a signed graph $G$ with a pair of distinguished edges $e_1, e_2 \in E(G)$ has the property that all cycles containing both $e_1$ and $e_2$ have the same sign. This answers a question of Zaslavsky