Nowhere-zero 4-flow in almost Petersen-minor free graphs

AbstractTutte [W.T. Tutte, On the algebraic theory of graph colorings, J. Combin. Theory 1 (1966) 15–20] conjectured that every bridgeless Petersen-minor free graph admits a nowhere-zero 4-flow. Let (P10)μ̄ be the graph obtained from the Petersen graph by contracting μ edges from a perfect matching. In this paper we prove that every bridgeless (P10)3̄-minor free graph admits a nowhere-zero 4-flow

Is the five-flow conjecture almost false?

The number of nowhere zero Z_Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh that \Phi_G(Q) has no real roots for Q \in (4,\infty) was recently disproved by Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q \in [5,\infty). We study the real roots of \Phi_G(Q) for a family of non-planar cubic graphs known as generalised Petersen graphs G(m,k). We show that the modified conjecture on real flow roots is also false, by exhibiting infinitely many real flow roots Q>5 within the class G(nk,k). In particular, we compute explicitly the flow polynomial of G(119,7), showing that it has real roots at Q\approx 5.0000197675 and Q\approx 5.1653424423. We moreover prove that the graph families G(6n,6) and G(7n,7) possess real flow roots that accumulate at Q=5 as n\to\infty (in the latter case from above and below); and that Q_c(7)\approx 5.2352605291 is an accumulation point of real zeros of the flow polynomials for G(7n,7) as n\to\infty.Comment: 44 pages (LaTeX2e). Includes tex file, three sty files, and a mathematica script polyG119_7.m. Many improvements from version 3, in particular Sections 3 and 4 have been mostly re-writen, and Sections 7 and 8 have been eliminated. (This material can now be found in arXiv:1303.5210.) Final version published in J. Combin. Theory

Cycle Double Covers and Integer Flows

My research focuses on two famous problems in graph theory, namely the cycle double cover conjecture and the integer flows conjectures. This kind of problem is undoubtedly one of the major catalysts in the tremendous development of graph theory. It was observed by Tutte that the Four color problem can be formulated in terms of integer flows, as well as cycle covers. Since then, the topics of integer flows and cycle covers have always been in the main line of graph theory research. This dissertation provides several partial results on these two classes of problems

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

Excluding Two Minors of the Petersen Graph

In this dissertation, we begin with a brief survey of the Petersen graph and its role in graph theory. We will then develop an alternative decomposition to clique sums for 3-connected graphs, called T-sums. This decomposition will be used in Chapter 2 to completely characterize those graphs which have no P_3 minor, where P_3 is a graph with 7 vertices, 12 edges, and is isomorphic to the graph created by contracting three edges of a perfect matching of the Petersen Graph. In Chapter 3, we determine the structure of any large internally 4-connected graph which has no P_2 minor, where P_2 is a graph on 8 vertices, 13 edges, and is isomorphic to the graph created by contracting two edges of a perfect matching of the Petersen Graph