Edge-coloring a graph GG so that every copy of a graph HH has an odd color class

Recently, Alon introduced the notion of an $H$-code for a graph $H$: a collection of graphs on vertex set $[n]$ is an $H$-code if it contains no two members whose symmetric difference is isomorphic to $H$. Let $D_{H}(n)$ denote the maximum possible cardinality of an $H$-code, and let $d_{H}(n)=D_{H}(n)/2^{n \choose 2}$. Alon observed that a lower bound on $d_{H}(n)$ can be obtained by attaining an upper bound on the number of colors needed to edge-color $K_n$ so that every copy of $H$ has an odd color class. Motivated by this observation, we define $g(G,H)$ to be the minimum number of colors needed to edge-color a graph $G$ so that every copy of $H$ has an odd color class. We prove $g(K_n,K_5) \le n^{o(1)}$ and $g(K_{n,n}, C_4) \le \frac{t}{2(t-1)}n+o(n)$ for all odd integers $t\geq 5$. The first result shows $d_{K_5}(n) \ge \frac{1}{n^{o(1)}}$ and was obtained independently in arXiv:2306.14682

Embedded graph 3-coloring and flows

A graph drawn in a surface is a near-quadrangulation if the sum of the lengths of the faces different from 4-faces is bounded by a fixed constant. We leverage duality between colorings and flows to design an efficient algorithm for 3-precoloring-extension in near-quadrangulations of orientable surfaces. Furthermore, we use this duality to strengthen previously known sufficient conditions for 3-colorability of triangle-free graphs drawn in orientable surfaces.Comment: 53 pages, 15 figure

Educational psychologists’ work with domestic violence and abuse: investigating the utility of a resource within casework practice

This study explores educational psychologists' (EPs') perceptions of practice with children and families who have experienced domestic violence and abuse (DVA). A literature review of DVA outcomes for children revealed negative impacts, alongside potential protective factors. Psychological theory for explaining these outcomes are reviewed, and whilst beneficial, each theory does not fully explain how the child develops. Bronfenbrenner's Process, Person, Context, Time theory (1989) was suggested as a valuable basis for creating supportive resources for EP practice, to be used during assessment, intervention and evaluation. Trialled in a case study of three EPs, semi-structured interviews were used before and after the use of the resource pack, to explore potential practice changes. Thematic analysis (Braun and Clarke, 2006) was employed to create themes, both deductively according to Bronfenbrenner's ecological categories, and driven by the data inductively. Analysis of ' pre-resource' practice suggested there were key aspects of Bronfenbrenner's bioecological model that were not commonly explored by the EPs. Moreover, barriers to practice were identified in the form of lack of time; limited DVA knowledge; and lack of appropriate resources. 'Post- resource' practice suggested many changes, which were perceived by the participants as providing them with a deeper understanding of the child. Overall it has been suggested that the resource pack has supported many practice developments, yet there are still areas which fail to be addressed. It is acknowledged that not all practice barriers will be overcome by a refined resource pack in isolation, yet participants claimed to find it a valuable addition to their practice 'toolkit'. As this resource pack is in its relative infancy, suggestions regarding its growth and development are made

Who Isn’t Biased? Perceived Bias as a Dimension of Credibility in Communication of Science with Policymakers

Identifying the determinants of success in communicating scientific information for use in policymaking has been a topic of longstanding interest for many fields. Perceptions of message and messenger credibility are recognized in many disciplines as key to information receptivity, and subsequent changes in attitudes, by decision-makers. Bias also appears as a component of credibility, but is not theoretically well-defined, and its effects can appear inconsistent. This study—comprised of interviews with congressional staff members with energy, environment, and science portfolios (n=16)—finds that perceptions of bias and expertise play a critical role in the way in which policymakers evaluate scientific information sources, building on early research on credibility and opinion change

Chronic morphine reduces surface expression of δ-opioid receptors in subregions of rostral striatum

The delta opioid receptor (DOPr), whilst not the primary target of clinically used opioids, is involved in development of opioid tolerance and addiction. There is growing evidence that DOPr trafficking is involved in drug addiction, e.g., a range of studies have shown increased plasma membrane DOPr insertion during chronic treatment with opioids. The present study used a transgenic mouse model in which the C-terminal of the DOPr is tagged with enhanced-green fluorescence protein to examine the effects of chronic morphine treatment on surface membrane expression in striatal cholinergic interneurons that are implicated in motivated learning following both chronic morphine and morphine sensitization treatment schedules in male mice. A sex difference was noted throughout the anterior striatum, which was most prominent in the nucleus accumbens core region. Incontrast with previous studies in other neurons, chronic exposure to a high dose of morphine for 6 days had no effect, or slightly decreased (anterior dorsolateral striatum) surface DOPr expression. A morphine sensitization schedule produced similar results with a significant decrease in surface DOPr expression in nucleus accumbens shell. These results suggest that chronic morphine and morphine sensitisation treatment may have effects on instrumental reward-seeking behaviours and learning processes related to drug addiction, via effects on striatal DOPr function.NHMRC Grants: 1045964 & 108925