Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees

A 1-factorization F of a complete graph K2n is said to be G-regular, or regular under G, if G is an automorphism group of F acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (Eur J Comb 6:45–48, 1985) on cyclic groups and it is still open when n is even, although several classes of groups were tested in the recent past. It has been recently proved, see Rinaldi (Australas J Comb 80(2):178–196, 2021) and Mazzuoccolo et al. (Discret Math 342(4):1006–1016, 2019), that a G-regular 1-factorization, together with a complete set of rainbow spanning trees, exists for each group G of order 2n, n odd. The existence for each even n>2 was proved when either G is cyclic and n is not a power of 2, or when G is a dihedral group. Explicit constructions were given in all these cases. In this paper we extend this result and give explicit constructions when n>2 is even and G is either abelian but not cyclic, dicyclic, or a non cyclic 2-group with a cyclic subgroup of index 2

Primitive collineation groups of ovals with a fixed point

AbstractWe investigate collineation groups of a finite projective plane of odd order n fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which the group fixes a point off the oval is considered. We prove that it occurs in a Desarguesian plane if and only if (n+1)/2 is an odd prime, the group lying in the normalizer of a Singer cycle of PGL(2,n) in this case. For an arbitrary plane we show that the group cannot contain Baer involutions and derive a number of structural and numerical properties in the case where the group has even order. The existence question for a non-Desarguesian example is addressed but remains unanswered, although such an example cannot have order n≤23 as computer searches carried out with GAP show

A class of complete arcs in multiply derived planes

We prove that unital-derived (q^2 - q + 1)-arcs of PG(2, q^2) still yield complete arcs after multiple derivation with respect to disjoint derivation sets on a given line

Beyond the limits of accountants as ‘trusted advisors’ for small and medium sized enterprises:environmental sustainability advice

We present emerging research on the potential of small accounting practices as advisors on environmental sustainability for their small and medium sized clients (SMEs). Enabling this resource could have substantial implications to improve sustainability because SMEs are notoriously difficult to reach and influence, but their most ‘trusted advisor’ is consistently their accountant. Using Habermas’ notion of steering, we present research data primarily from interviews with accountants which shows that the personal trust (relational steering mechanism) currently enjoyed is inadequate to facilitate the provision of sustainability advice by accountants beyond reflection on their own experiences. There is a lack of institutional trust in the profession as credible providers of sustainability advice, and individuals lack the skills and knowledge to offer environmental accounting. Without these, transactional steering and formalized environmental sustainability advice by small accountants to SMEs is unlikely to emerge. We conclude with research recommendations and implications for the accounting professio

k–Pyramidal One–Factorizations

We consider one–factorizations of complete graphs which possess an automorphism group fixing k ≥ 0 vertices and acting regularly (i.e., sharply transitively) on the others. Since the cases k = 0 and k = 1 are well known in literature, we study the case k>=2 in some detail. We prove that both k and the order of the group are even and the group necessarily contains k − 1 involutions. Constructions for some classes of groups are given. In particular we extend the result of [7]: let G be an abelian group of even order and with k − 1 involutions, a one–factorization of a complete graph admitting G as an automorphism group fixing k vertices and acting regularly on the others can be constructed

Sharply transitive 1-factorizations of complete multipartite graphs

Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic

Sharply transitive 1-factorizations of complete multipartite graphs

Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic

A non-existence result on cyclic cycle decomposition of the cocktail party graph

Non existence results on cyclic cycle decompoisitions of a complete graph with a 1-factor removed are given

Powerplan : The alternative software for presentations at local area network using virtualtreeview and omnithread

The need for LCD projectors is very urgent, so even though they often experience shortcomings, it is impossible to continue to make purchases when needed to repair LCD projectors that require high costs such as buying a new device. Software development from the results of this study is not only beneficial for educational institutions but also for personal or specific institutions and institutions who want to conduct seminars, presentations, or other activities that involve many audiences. The advantage is that the audience will not experience problems such as when using an LCD projector, such as limited visibility, location or space, and the viewpoint of the audience on the Projector LCD / Screen. Another contribution that can be given from this research is a model of the teaching and learning process in real-time without using an LCD and only using a PC/ Laptop connected to the same LAN. It is a development of research that supports the concept of green computing because it also reduces the use of equipment LCD projector, but the usage function remains the same even more effective and interactive even without the device. Presentation model like this is the development of new models in the learning process using LAN networks but without using the internet connection and is the cross-platform model. This research can be the answer to the need for using LCD projectors that often occur in the process of education, meetings, or seminars that continue to increase every time. Also, the lack of LCD compared to the number of teaching needs in each class and the unstable electricity conditions and the low electricity ratio in East Nusa Tenggara, have resulted in frequent blackouts suddenly, so that makes the device LCD vulnerable to damage. The test results showed an increase in RTT along with the increasing number of servers conducting presentation activities. Testing on LAN networks with the same number of clients on each test shows that there was an increase of 18% and 75% in each addition to the number of servers. The best performance of Powerplan is still shown when the number of servers that present several five, with the response time of each client connected under two seconds