Cyclically five-connected cubic graphs

A cubic graph $G$ is cyclically 5-connected if $G$ is simple, 3-connected, has at least 10 vertices and for every set $F$ of edges of size at most four, at most one component of $G\backslash F$ contains circuits. We prove that if $G$ and $H$ are cyclically 5-connected cubic graphs and $H$ topologically contains $G$, then either $G$ and $H$ are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph $G'$ such that $H$ topologically contains $G'$ and $G'$ is obtained from $G$ in one of the following two ways. Either $G'$ is obtained from $G$ by subdividing two distinct edges of $G$ and joining the two new vertices by an edge, or $G'$ is obtained from $G$ by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from $G$ in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of $H$ we are able to eliminate the second construction. We also prove versions of both of these results when $G$ is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case $G'$ is required to be almost cyclically 5-connected and to have fewer circuits of length four than $G$. In particular, if $G$ has at most one circuit of length four, then $G'$ is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs $G'$ are more complicated.Comment: 47 pages, 5 figures. Revised according to referee's comments. To appear in J. Combin. Theory Ser.

Excluded minors in cubic graphs

Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in G, and G is not one particular 20-vertex graph, then either G\v is planar for some vertex v, or G can be drawn with crossings in the plane, but with only two crossings, both on the infinite region. We also prove several other theorems of the same kind.Comment: 62 pages, 17 figure

Cyclically 5-Connected Graphs

Tutte's Four-Flow Conjecture states that every bridgeless, Petersen-free graph admits a nowhere-zero 4-flow. This hard conjecture has been open for over half a century with no significant progress in the first forty years. In the recent decades, Robertson, Thomas, Sanders and Seymour has proved the cubic version of this conjecture. Their strategy involved the study of the class of cyclically 5-connected cubic graphs. It turns out a minimum counterexample to the general Four-Flow Conjecture is also cyclically 5-connected. Motivated by this fact, we wish to find structural properties of this class in hopes of producing a list of minor-minimal cyclically 5-connected graphs

Modeling and Tuning of Energy Harvesting Device Using Piezoelectric Cantilever Array

Piezoelectric devices have been increasingly investigated as a means of converting ambient vibrations into electrical energy that can be stored and used to power other devices, such as the sensors/actuators, micro-electro-mechanical systems (MEMS) devices, and microprocessor units etc. The objective of this work was to design, fabricate, and test a piezoelectric device to harvest as much power as possible from vibration sources and effectively store the power in a battery.;The main factors determining the amount of collectable power of a single piezoelectric cantilever are its resonant frequency, operation mode and resistive load in the charging circuit. A proof mass was used to adjust the resonant frequency and operation mode of a piezoelectric cantilever by moving the mass along the cantilever. Due to the tiny amount of collected power, a capacitor was suggested in the charging circuit as an intermediate station. To harvest sufficient energy, a piezoelectric cantilever array, which integrates multiple cantilevers in parallel connection, was investigated.;In the past, most prior research has focused on the theoretical analysis of power generation instead of storing generated power in a physical device. In this research, a commercial solid-state battery was used to store the power collected by the proposed piezoelectric cantilever array. The time required to charge the battery up to 80% capacity using a constant power supply was 970 s. It took about 2400 s for the piezoelectric array to complete the same task. Other than harvesting energy from sinusoidal waveforms, a vibration source that emulates a real environment was also studied. In this research the response of a bridge-vehicle system was used as the vibration sources such a scenario is much closer to a real environment compared with typical lab setups

A conjetura dos 3-fluxos de Tutte e emparelhamentos em grafos bipartidos

Orientador : Ricardo DahabDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoMestrad

Three-edge-colouring doublecross cubic graphs

A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte [9] conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in general if it is true for apex graphs and doublecross graphs [6,7]. In another paper [8], two of us solved the apex case, but the doublecross case remained open. Here we solve the doublecross case; that is, we prove that every two-edge-connected doublecross cubic graph is three edge-colourable. The proof method is a variant on the proof of the four-colour theorem given in [5]. (C) 2015 Elsevier Inc. All rights reserved