On 22-cycles of graphs

Let $G=(V,E)$ be a finite undirected graph. Orient the edges of $G$ in an arbitrary way. A $2$-cycle on $G$ is a function $d : E^2\to \mathbb{Z}$ such for each edge $e$, $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$, and $d(e, f) = 0$ whenever $e$ and $f$ have a common vertex. We show that each $2$-cycle is a sum of three special types of $2$-cycles: cycle-pair $2$-cycles, Kuratowski $2$-cycles, and quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each $2$-cycle is a sum of cycle-pair $2$-cycles and at most one Kuratowski $2$-cycle. Furthermore, if $G$ is Kuratowski connected, we characterize when every Kuratowski $2$-cycle is a sum of cycle-pair $2$-cycles. A $2$-cycles $d$ on $G$ is skew-symmetric if $d(e,f) = -d(f,e)$ for all edges $e,f\in E$. We show that each $2$-cycle is a sum of two special types of skew-symmetric $2$-cycles: skew-symmetric cycle-pair $2$-cycles and skew-symmetric quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each skew-symmetric $2$-cycle is a sum of skew-symmetric cycle-pair $2$-cycles. Similar results like this had previously been obtained by one of the authors for symmetric $2$-cycles. Symmetric $2$-cycles are $2$-cycles $d$ such that $d(e,f)=d(f,e)$ for all edges $e,f\in E$

On Skew Braces (with an appendix by N. Byott and L. Vendramin)

Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.Comment: 37 pages. Final versio

Four-class Skew-symmetric Association Schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric amorphous association schemes, submitted for publication], we discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables and intersection numbers for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S. Y. Song [Commutative association schemes whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996]. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that no 2-class Johnson scheme can admit a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.Comment: 12 page

Decomposition of skew-morphisms of cyclic groups

A skew-morphism of a group ▫$H$▫ is a permutation ▫$sigma$▫ of its elements fixing the identity such that for every ▫$x, y in H$▫ there exists an integer ▫$k$▫ such that ▫$sigma (xy) = sigma (x)sigma k(y)$▫. It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups ▫$mathbb Z_n$▫: if ▫$n = n_{1}n_{2}$▫ such that ▫$(n_{1}n_{2}) = 1$▫, and ▫$(n_{1}, varphi (n_{2})) = (varphi (n_{1}), n_{2}) = 1$▫ (▫$varphi$▫ denotes Euler\u27s function) then all skew-morphisms ▫$sigma$▫ of ▫$mathbb Z_n$▫ are obtained as ▫$sigma = sigma_1 times sigma_2$▫, where ▫$sigma_i$▫ are skew-morphisms of ▫$mathbb Z_{n_i},i = 1, 2$▫. As a consequence we obtain the following result: All skew-morphisms of ▫$mathbb Z_n$▫ are automorphisms of ▫$mathbb Z_n$▫ if and only if ▫$n = 4$▫ or ▫$(n, varphi(n)) = 1$▫

Skew propagation time

The zero forcing number has long been used as a tool for determining the maximum nullity of a graph, and has since been extended to skew zero forcing number, which is the zero forcing number used when the matrices corresponding to the graphs in question are required to have zeros as their diagonal entries. Skew zero forcing is based on the following specific color change rule: for a graph $G$ (that does not contain any loops), where some vertices are colored blue and the rest of the vertices are colored white, if a vertex has only one white neighbor, that vertex forces its white neighbor to be blue. The minimum number of blue vertices that it takes to force the graph using this color change rule is called the skew zero forcing number. A set of blue vertices of order equal to the skew zero forcing number of the graph, such that when the skew zero forcing process is carried out to completion the entire graph is colored blue, is called a minimum skew zero forcing set. More recently, the concept of propagation time of a graph was introduced. Propagation time of a graph is how fast it is possible to force the entire graph blue over all possible minimum zero forcing sets. In this thesis, the concept of propagation time is extended to skew propagation time. We discuss the tools used to study extreme skew propagation time, and examine the skew propagation time of several common families of graphs. Finally, we include a brief discussion on loop graph propagation time, where the graphs in question are allowed to contain loops (in other words, these graphs are loop graphs, not simple graphs, as in the skew zero forcing case)