Fan-extensions in fragile matroids

If S is a set of matroids, then the matroid M is S-fragile if, for every element e in E(M), either M\e or M/e has no minor isomorphic to a member of S. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when F is a minor-closed class of S-fragile matroids, and N is in F, the only members of F that contain N as a minor are obtained from N by increasing the length of fans. We prove that if this is the case, then we can certify it with a finite case-analysis. The analysis involves examining matroids that are at most two elements larger than N.Comment: Small revisions and correction

The excluded minors for 2- and 3-regular matroids

The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of matroids representable over the Hydra-5 partial field, and the 3-connected matroids in the class with a $U_{2,5}$- or $U_{3,5}$-minor are precisely those with six inequivalent representations over GF(5). We also prove that an excluded minor for this class has at most 15 elements.Comment: 79 pages, 1 figur

Computing excluded minors for classes of matroids representable over partial fields

We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids

Fractional refinements of integral theorems

The focus of this thesis is to take theorems which deal with ``integral" objects in graph theory and consider fractional refinements of them to gain additional structure. A classic theorem of Hakimi says that for an integer $k$, a graph has maximum average degree at most $2k$ if and only if the graph decomposes into $k$ pseudoforests. To find a fractional refinement of this theorem, one simply needs to consider the instances where the maximum average degree is fractional. We prove that for any positive integers $k$ and $d$, if $G$ has maximum average degree at most $2k + \frac{2d}{k+d+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of pseudoforests has every connected component containing at most $d$ edges, and further this pseudoforest is acyclic. The maximum average degree bound is best possible for every choice of $k$ and $d$. Similar to Hakimi's Theorem, a classical theorem of Nash-Williams says that a graph has fractional arborcity at most $k$ if and only if $G$ decomposes into $k$ forests. The Nine Dragon Tree Theorem, proven by Jiang and Yang, provides a fractional refinement of Nash-Williams Theorem. It says, for any positive integers $k$ and $d$, if a graph $G$ has fractional arboricity at most $k + \frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests, where one of the forests has maximum degree $d$. We prove a strengthening of the Nine Dragon Tree Theorem in certain cases. Let $k=1$ and $d \in \{3,4\}$. Every graph with fractional arboricity at most $1 + \frac{d}{d+2}$ decomposes into two forests $T$ and $F$ where $F$ has maximum degree $d$, every component of $F$ contains at most one vertex of degree $d$, and if $d= 4$, then every component of $F$ contains at most $8$ edges $e=xy$ such that both $\deg(x) \geq 3$ and $\deg(y) \geq 3$. In fact, when $k = 1$ and $d=3$, we prove that every graph with fractional arboricity $1 + \frac{3}{5}$ decomposes into two forests $T,F$ such that $F$ has maximum degree $3$, every component of $F$ has at most one vertex of degree $3$, further if a component of $F$ has a vertex of degree $3$ then it has at most $14$ edges, and otherwise a component of $F$ has at most $13$ edges. Shifting focus to problems which partition the vertex set, circular colouring provides a way to fractionally refine colouring problems. A classic theorem of Tuza says that every graph with no cycles of length $1 \bmod k$ is $k$-colourable. Generalizing this to circular colouring, we get the following: Let $k$ and $d$ be relatively prime, with $k>2d$, and let $s$ be the element of $\mathbb{Z}_k$ such that $sd \equiv 1\mod k$. Let $xy$ be an edge in a graph $G$. If $G-xy$ is $(k,d)$-circular-colorable and $G$ is not, then $xy$ lies in at least one cycle in $G$ of length congruent to $is \mod k$ for some $i$ in $\{1,\ldots,d\}$. If this does not occur with $i \in\{1,\ldots,d-1\}$, then $xy$ lies in at least two cycles of length $1 \mod k$ and $G-xy$ contains a cycle of length $0 \mod k$. This theorem is best possible with regards to the number of congruence classes when $k = 2d+1$. A classic theorem of Gr\"{o}tzsch says that triangle free planar graphs are $3$-colourable. There are many generalizations of this result, however fitting the theme of fractional refinements, Jaeger conjectured that every planar graph of girth $4k$ admits a homomorphism to $C_{2k+1}$. While we make no progress on this conjecture directly, one way to approach the conjecture is to prove critical graphs have large average degree. On this front, we prove: Every $4$-critical graph which does not have a $(7,2)$-colouring and is not $K_{4}$ or $W_{5}$ satisfies $e(G) \geq \frac{17v(G)}{10}$, and every triangle free $4$-critical graph satisfies $e(G) \geq \frac{5v(G)+2}{3}$. In the case of the second theorem, a result of Davies shows there exists infinitely many triangle free $4$-critical graphs satisfying $e(G) = \frac{5v(G) +4}{3}$, and hence the second theorem is close to being tight. It also generalizes results of Thomas and Walls, and also Thomassen, that girth $5$ graphs embeddable on the torus, projective plane, or Klein bottle are $3$-colourable. Lastly, a theorem of Cereceda, Johnson, and van den Heuvel, says that given a $2$-connected bipartite planar graph $G$ with no separating four-cycles and a $3$-colouring $f$, then one can obtain all $3$-colourings from $f$ by changing one vertices' colour at a time if and only if $G$ has at most one face of size $6$. We give the natural generalization of this to circular colourings when $\frac{p}{q} < 4$

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Atomistic modeling of the charge process and optimization of catalysts positioning in porous cathodes of lithium/air batteries

The reversibility and capacity of current lithium/air cells are severely limited by the high overpotential between the charge and discharge process and the occlusion of the pores of the active cathode surface due to non-uniform deposition of Li2O2 as the discharge product. In this thesis we present a study of these capacity-limiting issues on the lithium/air battery in two parts. First we present a combined classical and density functional theory based molecular dynamics study of the mechanisms underlying the oxygen evolution reaction during the charging of lithium/air batteries. As models for the Li2O2 material at the cathode we employ small amorphous clusters with a 2:2 Li:O stoichiometry, whose energetically most stable atomic configurations comprise both O atoms and O-O pairs with mixed peroxide/superoxide character, as revealed by their bond lengths, charges, spin moments, and densities of states. The oxidation of Li8O8 clusters is studied in unbiased density functional theory based molecular dynamics simulations upon removal of either one or two electrons, either in vacuo or immersed in dimethyl sulfoxide solvent molecules with a structure previously optimized by means of classical molecular dynamics. Whereas removal of one electron leads only to an enhancement of the superoxide character of O-O bonds, removal of two electrons leads to the spontaneous dissolution of either an O2 or a LiO2 molecule. These results are interpreted in terms of a two-stage process in which a peroxide-to-superoxide transition can take place in amorphous Li2O2 phases at low oxidation potentials, later followed by the dissolution of dioxygen molecules and Li ions at higher potentials. In the second part we solve numerically a reaction-diffusion equation to determine the Li2O2 deposition profiles in a model porous cathode in the absence and presence of discrete catalytic sites, considering four commonly used electrolytes. We implement a Greedy optimization algorithm to maximize the cathode capacity before pore clogging by optimal positioning of the discrete catalysts along the pore. The results indicate that a maximal capacity is limited by the oxygen solubility and diffusivity in each electrolyte in the absence of catalysts and vary widely in the four cases considered. However, optimal catalyst distributions can effectively compensate for these differences, suggesting a rational way of designing cathode structures with high performances according to the required operation conditions

Risk-averse multi-armed bandits and game theory

The multi-armed bandit (MAB) and game theory literature is mainly focused on the expected cumulative reward and the expected payoffs in a game, respectively. In contrast, the rewards and the payoffs are often random variables whose expected values only capture a vague idea of the overall distribution. The focus of this dissertation is to study the fundamental limits of the existing bandits and game theory problems in a risk-averse framework and propose new ideas that address the shortcomings. The author believes that human beings are mostly risk-averse, so studying multi-armed bandits and game theory from the point of view of risk aversion, rather than expected reward/payoff, better captures reality. In this manner, a specific class of multi-armed bandits, called explore-then-commit bandits, and stochastic games are studied in this dissertation, which are based on the notion of Risk-Averse Best Action Decision with Incomplete Information (R-ABADI, Abadi is the maiden name of the author's mother). The goal of the classical multi-armed bandits is to exploit the arm with the maximum score defined as the expected value of the arm reward. Instead, we propose a new definition of score that is derived from the joint distribution of all arm rewards and captures the reward of an arm relative to those of all other arms. We use a similar idea for games and propose a risk-averse R-ABADI equilibrium in game theory that is possibly different from the Nash equilibrium. The payoff distributions are taken into account to derive the risk-averse equilibrium, while the expected payoffs are used to find the Nash equilibrium. The fundamental properties of games, e.g. pure and mixed risk-averse R-ABADI equilibrium and strict dominance, are studied in the new framework and the results are expanded to finite-time games. Furthermore, the stochastic congestion games are studied from a risk-averse perspective and three classes of equilibria are proposed for such games. It is shown by examples that the risk-averse behavior of travelers in a stochastic congestion game can improve the price of anarchy in Pigou and Braess networks. Furthermore, the Braess paradox does not occur to the extent proposed originally when travelers are risk-averse. We also study an online affinity scheduling problem with no prior knowledge of the task arrival rates and processing rates of different task types on different servers. We propose the Blind GB-PANDAS algorithm that utilizes an exploration-exploitation scheme to load balance incoming tasks on servers in an online fashion. We prove that Blind GB-PANDAS is throughput optimal, i.e. it stabilizes the system as long as the task arrival rates are inside the capacity region. The Blind GB-PANDAS algorithm is compared to FCFS, Max-Weight, and c-mu-rule algorithms in terms of average task completion time through simulations, where the same exploration-exploitation approach as Blind GB-PANDAS is used for Max-Weight and c-$\mu$-rule. The extensive simulations show that the Blind GB-PANDAS algorithm conspicuously outperforms the three other algorithms at high loads