Hamiltonicity of 3-arc graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of $G$. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are Hamiltonian. As a consequence we obtain that if a vertex-transitive graph is isomorphic to the 3-arc graph of a connected arc-transitive graph of degree at least three, then it is Hamiltonian. This confirms the well known conjecture, that all vertex-transitive graphs with finitely many exceptions are Hamiltonian, for a large family of vertex-transitive graphs. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative 3-arc graphs.Comment: in press Graphs and Combinatorics, 201

Hawaii coastal seawater CO2 network: A statistical evaluation of a decade of observations on tropical coral reefs.

© The Author(s), 2019. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Terlouw, G. J., Knor, L. A. C. M., De Carlo, E. H., Drupp, P. S., Mackenzie, F. T., Li, Y. H., Sutton, A. J., Plueddemann, A. J., & Sabine, C. L. Hawaii coastal seawater CO2 network: A statistical evaluation of a decade of observations on tropical coral reefs. Frontiers in Marine Science, 6, (2019):226, doi:10.3389/fmars.2019.00226.A statistical evaluation of nearly 10 years of high-resolution surface seawater carbon dioxide partial pressure (pCO2) time-series data collected from coastal moorings around O’ahu, Hawai’i suggest that these coral reef ecosystems were largely a net source of CO2 to the atmosphere between 2008 and 2016. The largest air-sea flux (1.24 ± 0.33 mol m−2 yr−1) and the largest variability in seawater pCO2 (950 μatm overall range or 8x the open ocean range) were observed at the CRIMP-2 site, near a shallow barrier coral reef system in Kaneohe Bay O’ahu. Two south shore sites, Kilo Nalu and Ala Wai, also exhibited about twice the surface water pCO2 variability of the open ocean, but had net fluxes that were much closer to the open ocean than the strongly calcifying system at CRIMP-2. All mooring sites showed the opposite seasonal cycle from the atmosphere, with the highest values in the summer and lower values in the winter. Average coastal diurnal variabilities ranged from a high of 192 μatm/day to a low of 32 μatm/day at the CRIMP-2 and Kilo Nalu sites, respectively, which is one to two orders of magnitude greater than observed at the open ocean site. Here we examine the modes and drivers of variability at the different coastal sites. Although daily to seasonal variations in pCO2 and air-sea CO2 fluxes are strongly affected by localized processes, basin-scale climate oscillations also affect the variability on interannual time scales.We acknowledge with gratitude the financial support of our research provided in part by a grant/cooperative agreement from the National Oceanic and Atmospheric Administration, Project R/IR-27, which is sponsored by the University of Hawaii Sea Grant College Program, SOEST, under Institutional Grant No. NA14OAR4170071 from NOAA Office of Sea Grant, Department of Commerce. Additional support was granted by the NOAA/Ocean Acidification Program (to EDC and AS) and the NOAA/Climate Program Office (AP), and the NOAA Ocean Observing and Monitoring Division, Climate Program Office (FundRef number 100007298) through agreement NA14OAR4320158 of the NOAA Cooperative Institute for the North Atlantic Region (AP). The views expressed herein are those of the author(s) and do not necessarily reflect the views of NOAA or any of its subagencies. This is SOEST contribution number 10684, PMEL contribution number 4845, and Hawai’i Sea Grant contribution UNIHI-SEAGRANT-JC-15-30

Rigid Steiner triple systems obtained from projective triple systems

It was shown by Babai in 1980 that almost all Steiner triple systems are rigid; that is, their only automorphism is the identity permutation. Those Steiner triple systems with the largest automorphism groups are the projective systems of orders $2^n-1$. In this paper we show that each such projective system may be transformed to a rigid Steiner triple system by at most $n$ Pasch trades whenever $n\ge 4$

Dihedral biembeddings and triangulations by complete and complete tripartite graphs

We construct biembeddings of some Latin squares which are Cayley tables of dihedral groups. These facilitate the construction of $n^{an^2}$ nonisomorphic face 2-colourable triangular embeddings of the complete tripartite graph $K_{n,n,n}$ and the complete graph $K_n$ for linear classes of values of $n$ and suitable constants $a$. Previously the best known lower bounds for the number of such embeddings that are applicable to linear classes of values of $n$ were of the form $2^{an^2}$. We remark that trivial upper bounds are $n^{n^2/3}$ in the case of complete graphs $K_n$ and $n^{2n^2}$ in the case of complete tripartite graphs $K_{n,n,n}$

Biembeddings of Latin squares of side 8

Face 2-colourable triangular embeddings of complete tripartite graphs $K_{n,n,n}$ correspond to biembeddings of Latin squares of side $n$. We consider biembeddings that contain any of the five Latin squares derived from the Cayley tables of finite groups of order 8. Up to isomorphism, we determine all such biembeddings