On flows of graphs

Tutte\u27s 3-flow Conjecture, 4-flow Conjecture, and 5-flow Conjecture are among the most fascinating problems in graph theory. In this dissertation, we mainly focus on the nowhere-zero integer flow of graphs, the circular flow of graphs and the bidirected flow of graphs. We confirm Tutte\u27s 3-flow Conjecture for the family of squares of graphs and the family of triangularly connected graphs. In fact, we obtain much stronger results on this conjecture in terms of group connectivity and get the complete characterization of such graphs in those families which do not admit nowhere-zero 3-flows. For the circular flows of graphs, we establish some sufficient conditions for a graph to have circular flow index less than 4, which generalizes a new known result to a large family of graphs. For the Bidirected Flow Conjecture, we prove it to be true for 6-edge connected graphs

Integer flows and cycle covers

AbstractResults related to integer flows and cycle covers are presented. A cycle cover of a graph G is a collection C of cycles of G which covers all edges of G; C is called a cycle m-cover of G if each edge of G is covered exactly m times by the members of C. By using Seymour's nowhere-zero 6-flow theorem, we prove that every bridgeless graph has a cycle 6-cover associated to covering of the edges by 10 even subgraphs (an even graph is one in which each vertex is of even degree). This result together with the cycle 4-cover theorem implies that every bridgeless graph has a cycle m-cover for any even number m ≥ 4. We also prove that every graph with a nowhere-zero 4-flow has a cycle cover C such that the sum of lengths of the cycles in C is at most |E(G)| + |V(G)| − 2, unless G belongs to a very special class of graphs

Combinatorially Thin Trees and Spectrally Thin Trees in Structured Graphs

Given a graph $G=(V,E)$, finding simpler estimates of $G$ with possibly fewer edges or vertices while capturing some of its specific properties has been used in order to design efficient algorithms. The concept of estimating a graph with a simpler graph is known as graph sparsification. Spanning trees are an important family of graph sparsifiers that maintain connectivity of graphs, and have been utilized in many applications. However, spanning trees are a wide family, and for some applications one might need the spanning tree to have specific properties. Combinatorially thin trees are a type of spanning trees that show up in applications such as Asymmetric Travelling Salesman Problem (ATSP). A spanning tree $T$ of $G$ is combinatorially thin if there is no cut $U\subset V$ such that $T$ contains all the edges in $\delta(U)$, and the thinness parameter $\alpha_G(T)$ measures the maximum fraction of edges in $E(T)\cap \delta(U)$ compared to $\delta(U)$ over all cuts $U\subset V$. Intuitively, combinatorial thinness measures how much edge-connectivity we lose while removing the spanning tree $T$ from $G$. It is easy to verify that if $G$ has connectivity $k$, then $\frac{1}{k}$ lower bounds $\alpha_G$. On the other hand, Goddyn conjectured that $\alpha_G$ can also be upper bounded as a function of connectivity $\alpha_G = f(\frac{1}{k})$. This conjecture which is known as thin tree conjecture, was proved for the special case of graphs with bounded genus by Oveis-Gharan and Saberi, in 2011. However, the general case is still open. In the first part of this thesis, we study some of the known connections between edge-connectivity and $\alpha_{G}$ and investigate the result of Oveis-Gharan and Saberi for the special case of planar graphs. For a general graph $G$ and spanning tree $T$, even verifying the combinatorial thinness $\alpha_{G}(T)$ of $T$ is an $\text{NP}$-hard problem. A natural more efficiently computable relaxation of combinatorial thinness is the notion of spectral thinness. For a graph $G$ and a spanning tree $T$ in $G$ the spectral thinness $\theta_{G}(T)$ is the smallest value of $\theta$ such that \theta\L_G - \L_T is a positive semidefinite matrix where \L_G and \L_T are Laplacian matrices of $G$ and $T$. Additionally, we define $\theta_G$ to be the minimum value of $\theta_{G}(T)$ over all spanning trees $T$ of $G$. Similar to combinatorial thinness and connectivity, $\theta_{G}(T)$ can be lower bounded by the maximum effective resistance of edges in $T$. It was also proven by Harvey and Olver in 2014 that the maximum effective resistance of edges in $G$ asymptotically upper bounds $\theta_{G}$. However, finding a mathematical characterization of $\theta_{G}(T)$, even for structured graphs, is still a challenge. In the second part of this thesis, we will give general lower bound and upper bound certificates for $\theta_{G}(T)$ and utilize these certificates for circulant matrices to estimate spectral thinness of graphs such as complete graphs, complete bipartite graphs, and prism graphs

Graph Theory

This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results

Integer Flows and Circuit Covers of Graphs and Signed Graphs

The work in Chapter 2 is motivated by Tutte and Jaeger\u27s pioneering work on converting modulo flows into integer-valued flows for ordinary graphs. For a signed graphs (G, sigma), we first prove that for each k ∈ {lcub}2, 3{rcub}, if (G, sigma) is (k -- 1)-edge-connected and contains an even number of negative edges when k = 2, then every modulo k-flow of (G, sigma) can be converted into an integer-valued ( k + 1)-ow with a larger or the same support. We also prove that if (G, sigma) is odd-(2p+1)-edge-connected, then (G, sigma) admits a modulo circular (2 + 1/ p)-flows if and only if it admits an integer-valued circular (2 + 1/p)-flows, which improves all previous result by Xu and Zhang (DM2005), Schubert and Steffen (EJC2015), and Zhu (JCTB2015).;Shortest circuit cover conjecture is one of the major open problems in graph theory. It states that every bridgeless graph G contains a set of circuits F such that each edge is contained in at least one member of F and the length of F is at most 7/5∥E(G)∥. This concept was recently generalized to signed graphs by Macajova et al. (JGT2015). In Chapter 3, we improve their upper bound from 11∥E( G)∥ to 14/3 ∥E(G)∥, and if G is 2-edgeconnected and has even negativeness, then it can be further reduced to 11/3 ∥E(G)∥.;Tutte\u27s 3-flow conjecture has been studied by many graph theorists in the last several decades. As a new approach to this conjecture, DeVos and Thomassen considered the vectors as ow values and found that there is a close relation between vector S1-flows and integer 3-NZFs. Motivated by their observation, in Chapter 4, we prove that if a graph G admits a vector S1-flow with rank at most two, then G admits an integer 3-NZF.;The concept of even factors is highly related to the famous Four Color Theorem. We conclude this dissertation in Chapter 5 with an improvement of a recent result by Chen and Fan (JCTB2016) on the upperbound of even factors. We show that if a graph G contains an even factor, then it contains an even factor H with.;∥E(H)∥ ≥ 4/7 (∥ E(G)∥+1)+ 1/7 ∥V2 (G)∥, where V2( G) is the set of vertices of degree two

Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of $T$, then $G$ can be edge-decomposed into subgraphs isomorphic to $T$. So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove a weaker version of the Tree Decomposition Conjecture, where we require the subgraphs in the decomposition to be isomorphic to graphs that can be obtained from $T$ by vertex-identifications. We call such a subgraph a homomorphic copy of $T$. This implies the Tree Decomposition Conjecture under the additional constraint that the girth of $G$ is greater than the diameter of $T$. As an application, we verify the Tree Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page