Vertex Partitions and Maximum \G-free Subgraphs

We define a $(V_1, V_2, \ldots, V_k)$-partition for a given graph $H$ and graphical properties $P_1, P_2, \ldots, P_k$ as a partition where each $V_i$ induces a subgraph of $H$ with property $P_i$. Matamala (2007) extended this result by showing that for any graph $H$ with $\Delta(H)=p+q$, there exists a $(V_1, V_2)$-partition of $V(H)$ where $H[V_1]$ is a maximum order $(p-1)$-degenerate induced subgraph and $H[V_2]$ is $(q-1)$-degenerate. Additionally, Catlin and Lai proved that if $\Delta(H)\geq 5$, $H$ has a $(V_1, V_2)$-partition such that $H[V_1]$ is a maximum order acyclic induced subgraph, $\omega(H[V_2])\leq \Delta(H)-2$, and $\Delta(H[V_2])\leq \Delta(H)-2$. Rowshan and Taherkhani demonstrated that given a graph $G$ with a minimum degree $\delta(G)$ and for $k=\lceil \frac{\Delta(H)}{\delta(G)}\rceil$, there exists a $(V_1, V_2, \ldots, V_k)$-partition of the vertex set of $H$, such that each $H[V_i]$ is $G$-free, meaning it does not contain a subgraph isomorphic to $G$, and $H[V_1]$ is a maximum order $G$-free induced subgraph. In our paper, we present a novel result for a connected graph $H$ with $\Delta(H)\geq 5$ and without $K_{\Delta(H)+1}\setminus e$ as a subgraph. We establish that when $p_1\geq p_2\geq\cdots\geq p_{k-1}\geq 2$, $p_k\geq 4$, $\sum_{i=1}^k p_i=\Delta(H)-1+k$, and $\mathcal{G}_i$ represents a family of graphs with a minimum degree at least $p_i-1$ for each $i\in [k-1]$, a $(V_1, V_2, \ldots, V_k)$-partition of $V(H)$ exists. This partition guarantees that $H[V_1]$ is a maximum order $\mathcal{G}_1$-free induced subgraph, $H[V_i]$ is $\mathcal{G}_i$-free for each $2\leq i\leq k-1$, $\Delta(H[V_k])\leq p_k$, and either $H[V_k]$ is $K_{p_k}$-free or its $p_k$-cliques are disjoint

Diversity of Methylobacterium Species Associated with New Zealand Native Plants

The genus Methylobacterium are pink-pigmented facultative methylotrophs (PPFMs), and are abundant colonizers of the phyllosphere, due to the availability of methanol, a waste product of pectin metabolism during plant cell division. Besides methanol cycling, Methylobacterium has important effects on plant health. The phyllosphere is an extreme environment with a landscape that is heterogeneous, continuously changing as the plant grows, and is exposed to very high ultra violet irradiation. Geographically, New Zealand has been isolated for over a million years, has a biologically diverse group of species, and is considered a biodiversity hotspot, with most of the native plants being endemic. We hypothesize that NZ native plants harbor diverse groups of Methylobacterium species, and to test this we aimed to isolate Methylobacterium species from the phyllosphere of native New Zealand plants. A leaf imprinting technique using methanol supplemented AMS agar media was used to isolate bacteria and diversity was determined using a combination of ARDRA, 16S rRNA gene sequencing and phylogenetic analysis. Methylobacterium spp. were successfully isolated from 19 of the 21 plant species used in this study. Eleven Methylobacterium species have been identified in association with the phyllosphere of native NZ plants: M. adhaesivum, M. brachiatum, M. komagatae, M. marchantiae, M. mesophilicum, M. phyllosphaerae, M. fujisawaense, M. oryzae, M. radiotolerans, M. tardum and M. zatmanii, with the first six being the most frequently isolated from more plant species. In this study other α, β, γ-proteobacterial species were also isolated: Hyphomicrobium, Methylopila, Rhizobium, Achromobacter, Methylophilus, Ramlibacter and Xanthomonas; Janibacter melonis (Actinomycetes); Niastella populi (Bacteroidetes) and Paenibacillus lautus (Firmicutes), highlighting the presence of potential novel methanol utilizer within the ecosystem. Results from this study indicate that Methylobacterium are abundant and dominant members of the NZ phyllosphere environment, with species diversity and composition dependent on the host plant species

Borodin-Kostochka conjecture and Partitioning a graph into classes with no clique of specified size

For a given graph $H$ and the graphical properties $P_1, P_2,\ldots,P_k$, a graph $H$ is said to be $(V_1, V_2,\ldots,V_k)$-partitionable if there exists a partition of $V(H)$ into $k$-sets $V_1, V_2\ldots,V_k$, such that for each $i\in[k]$, the subgraph induced by $V_i$ has the property $P_i$. In $1979$, Bollob\'{a}s and Manvel showed that for a graph $H$ with maximum degree $\Delta(H)\geq 3$ and clique number $\omega(H)\leq \Delta(H)$, if $\Delta(H)= p+q$, then there exists a $(V_1,V_2)$-partition of $V(H)$, such that $\Delta(H[V_1])\leq p$, $\Delta(H[V_2])\leq q$, $H[V_1]$ is $(p-1)$-degenerate, and $H[V_2]$ is $(q-1)$-degenerate. Assume that $p_1\geq p_2\geq\cdots\geq p_k\geq 2$ are $k$ positive integers and $\sum_{i=1}^k p_i=\Delta(H)-1+k$. Assume that for each $i\in[k]$ the properties $P_i$ means that $\omega(H[V_i])\leq p_i-1$. Is $H$ a $(V_1,\ldots,V_k)$-partitionable graph? In 1977, Borodin and Kostochka conjectured that any graph $H$ with maximum degree $\Delta(H)\geq 9$ and without $K_{\Delta(H)}$ as a subgraph, has chromatic number at most $\Delta(H)-1$. Reed proved that the conjecture holds whenever $\Delta(G) \geq 10^{14}$. When $p_1=2$ and $\Delta(H)\geq 9$, the above question is the Borodin and Kostochka conjecture. Therefore, when all $p_i$s are equal to $2$ and $\Delta(H)\leq 8$, the answer to the above question is negative. Let $H$ is a graph with maximum degree $\Delta$, and clique number $\omega(H)$, where $\omega(H)\leq \Delta-1$. In this article, we intend to study this question when $k\geq 2$ and $\Delta\geq 13$. In particular as an analogue of the Borodin-Kostochka conjecture, for the case that $\Delta\geq 13$ and $p_i\geq 2$ we prove that the above question is true

Segmented GRAND: Combining Sub-patterns in Near-ML Order

The recently introduced maximum-likelihood (ML) decoding scheme called guessing random additive noise decoding (GRAND) has demonstrated a remarkably low time complexity in high signal-to-noise ratio (SNR) regimes. However, the complexity is not as low at low SNR regimes and low code rates. To mitigate this concern, we propose a scheme for a near-ML variant of GRAND called ordered reliability bits GRAND (or ORBGRAND), which divides codewords into segments based on the properties of the underlying code, generates sub-patterns for each segment consistent with the syndrome (thus reducing the number of inconsistent error patterns generated), and combines them in a near-ML order using two-level integer partitions of logistic weight. The numerical evaluation demonstrates that the proposed scheme, called segmented ORBGRAND, significantly reduces the average number of queries at any SNR regime. Moreover, the segmented ORBGRAND with abandonment also improves the error correction performance

The bipartite Ramsey numbers BR(C8,C2n)BR(C_8, C_{2n})

For the given bipartite graphs $G_1,G_2,\ldots,G_t$, the multicolor bipartite Ramsey number $BR(G_1,G_2,\ldots,G_t)$ is the smallest positive integer $b$ such that any $t$-edge-coloring of $K_{b,b}$ contains a monochromatic subgraph isomorphic to $G_i$, colored with the $i$th color for some $1\leq i\leq t$. We compute the exact values of the bipartite Ramsey numbers $BR(C_8,C_{2n})$ for $n\geq2$

FORMULATION AND IN-VITRO/EX-VIVO CHARACTERIZATIONS OF MICROEMULSION-BASED HYDROGEL FORMULATION OF ACECLOFENAC FOR TOPICAL APPLICATION

ABSTRACTObjectives: In this study, microemulsion-based hydrogel (MBH) formulation of aceclofenac was prepared for topical administration in the managementof pain and inflammation to overcome the gastrointestinal toxicity of the drug occurs with oral administration.Methods: The MBH formulation was prepared by two-step methods. In the first step, the drug loaded o/w microemulsion was prepared first bytitration method and in the second step; xanthan gum was added slowly to the microemulsion under homogenization to produce clear MBH. Thedeveloped MBH formulations were characterized by in-vitro evaluations, stability studies, and skin irritancy test. The ex-vivo permeation across ratepidermis using modified Keshary-Chien diffusion cell and anti-inflammatory activity of the selected MBH formulation was also evaluated in rat hindpaw edema model.Results: The developed MBH formulations showed good stability and acceptable physicochemical properties. The selected formulation (MBH2)showed the highest skin permeation rate (transdermal flux, 193.59±5.01 µg/cm/h; lag time, 0.80±0.01 h) and a maximum of 70.96% inhibition ofthe hind paw edema was measured after 8 h of the study.2Conclusion: Thus, the results obtained in this study suggest the feasibility of the MBH formulation of aceclofenac for topical application for thetreatment of pain and inflammation.Keywords: Aceclofenac, Microemulsion-based hydrogel, Transdermal flux, Ex-vivo permeation