Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying $2$-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.Comment: 33 pages; submitted for publicatio

On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure

Light thresholds for seagrasses of the GBRWHA: a synthesis and guiding document. Including knowledge gaps and future priorities

[Extract] This synthesis contains light thresholds for seagrass species in the Great Barrier Reef World Heritage Area (GBRWHA). The thresholds can be applied to ensure protection of seagrasses from activities that impact water quality and the light environment over the short-term, such as coastal and port developments. Thresholds for long-term maintenance of seagrasses are also proposed

Gully Formation at the Haughton Impact Structure (Arctic Canada) Through the Melting of Snow and Ground Ice, with Implications for Gully Formation on Mars

The formation of gullies on Mars has been the topic of active debate and scientific study since their first discovery by Malin and Edgett in 2000. Several mechanisms have been proposed to account for gully formation on Mars, from dry mass movement processes, release of water or brine from subsurface aquifers, and the melting of near-surface ground ice or snowpacks. In their global documentation of martian gullies, report that gullies are confined to ~2783S and ~2872N latitudes and span all longitudes. Gullies on Mars have been documented on impact crater walls and central uplifts, isolated massifs, and on canyon walls, with crater walls being the most common situation. In order to better understand gully formation on Mars, we have been conducting field studies in the Canadian High Arctic over the past several summers, most recently in summer 2018 and 2019 under the auspices of the Canadian Space Agency-funded Icy Mars Analogue Program. It is notable that the majority of previous studies in the Arctic and Antarctica, including our recent work on Devon Island, have focused on gullies formed on slopes generated by regular endogenic geological processes and in regular bedrock. How-ever, as noted above, meteorite impact craters are the most dominant setting for gullies on Mars. Impact craters provide an environment with diverse lithologies including impact-generated and impact-modified rocks and slope angle, and thus greatly variable hill slope processes could occur within a localized area. Here, we investigate the formation of gullies within the Haughton impact structure and compare them to gullies formed in unimpacted target rock in the nearby Thomas Lee Inle

Observations of the high vibrational levels of the B′′B̄ 1Σ+u state of H2

Double-resonance laser spectroscopy via the E F 1Σ+g, v\u27 = 6, J\u27 = 0–2 state was used to probe the high vibrational levels of the B′′B̄ 1Σ+u state of molecular hydrogen. Resonantly enhanced multiphotonionization spectra were recorded by detecting ion production as a function of energy using a time of flight mass spectrometer. New measurements of energies for the v = 51–66 levels for the B00B ̄ state of H2 are reported, which, taken with previous results, span the v = 46–69 vibrational levels. Results for energy levels are compared to theoretical close-coupled calculations [L. Wolniewicz, T. Orlikowski, and G. Staszewska, J. Mol. Spectrosc. 238, 118–126 (2006)]. The average difference between the 84 measured energies and calculated energies is 3.8 cm–1 with a standard deviation of 5.3 cm–1. This level of agreement showcases the success of the theoretical calculations in accounting for the strong rovibronic mixing of the 1Σu+ and 1Πu+ states. Due to the ion-pair character of the outer well, the observed energies of the vibrational levels below the third dissociation limit smoothly connect with previously observed energies of ion-pair states above this limit. The results provide an opportunity for testing a heavy Rydberg multi-channel quantum defect analysis of the high vibrational states below the third dissociation limit

Observations of the high vibrational levels of the B′′B̄ 1Σ+u state of H2

Double-resonance laser spectroscopy via the E F 1Σ+g, v\u27 = 6, J\u27 = 0–2 state was used to probe the high vibrational levels of the B′′B̄ 1Σ+u state of molecular hydrogen. Resonantly enhanced multiphotonionization spectra were recorded by detecting ion production as a function of energy using a time of flight mass spectrometer. New measurements of energies for the v = 51–66 levels for the B00B ̄ state of H2 are reported, which, taken with previous results, span the v = 46–69 vibrational levels. Results for energy levels are compared to theoretical close-coupled calculations [L. Wolniewicz, T. Orlikowski, and G. Staszewska, J. Mol. Spectrosc. 238, 118–126 (2006)]. The average difference between the 84 measured energies and calculated energies is 3.8 cm–1 with a standard deviation of 5.3 cm–1. This level of agreement showcases the success of the theoretical calculations in accounting for the strong rovibronic mixing of the 1Σu+ and 1Πu+ states. Due to the ion-pair character of the outer well, the observed energies of the vibrational levels below the third dissociation limit smoothly connect with previously observed energies of ion-pair states above this limit. The results provide an opportunity for testing a heavy Rydberg multi-channel quantum defect analysis of the high vibrational states below the third dissociation limit

Coupling between downstream variations of channel width and local pool–riffle bed topography

A potential control of downstream channel width variations on the structure and planform of pool–riffle sequence local bed topography is a key to the dynamics of gravel bed rivers. How established pool–riffle sequences respond to time-varying changes in channel width at specific locations, however, is largely unexplored and challenging to address with field-based study. Here, we report results of a flume experiment aimed at building understanding of how statistically steady pool–riffle sequence profiles adjust to spatially prescribed channel width changes. We find that local bed slopes near steady-state conditions inversely correlate with local downstream width gradients when the upstream sediment supply approximates the estimated transport capacity. This result constrains conditions prior to and following the imposed local width changes. Furthermore, this relationship between local channel bed slope and downstream width gradient is consistent with expectations from scaling theory and a broad set of field-based, numerical, and experimental studies (n=88). However, upstream disruptions to coarse sediment supply through actions such as dam removal can result in a transient flipping of the expected inverse correlation between bed slope and width gradient, collectively highlighting that understanding local conditions is critical before typically implemented spatial averaging schemes can be reliably applied.Postprint (published version

Spectral Measures of Bipartivity in Complex Networks

We introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network. Spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity. It is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks, trophic interactions in food webs, construction principles in metabolic networks, or communities in social networks.Comment: 16 pages, 1 figure, 1 tabl

Divergence in Dialogue

Copyright: 2014 Healey et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.This work was supported by the Economic and Social Research Council (ESRC; http://www.esrc.ac.uk/) through the DynDial project (Dynamics of Conversational Dialogue, RES-062-23-0962) and the Engineering and Physical Sciences Research Council (EPSRC; http://www.epsrc.ac.uk/) through the RISER project (Robust Incremental Semantic Resources for Dialogue, EP/J010383/1). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript