A Characterization of Graphs with Small Palette Index

Given an edge-coloring of a graph G, we associate to every vertex v of G the set of colors appearing on the edges incident with v. The palette index of G is defined as the minimum number of such distinct sets, taken over all possible edge-colorings of G. A graph with a small palette index admits an edge-coloring which can be locally considered to be almost symmetric, since few different sets of colors appear around its vertices. Graphs with palette index 1 are r-regular graphs admitting an r-edge-coloring, while regular graphs with palette index 2 do not exist. Here, we characterize all graphs with palette index either 2 or 3 in terms of the existence of suitable decompositions in regular subgraphs. As a corollary, we obtain a complete characterization of regular graphs with palette index 3

A sharp upper bound for the harmonious total chromatic number of graphs and multigraphs

A proper total colouring of a graph $G$ is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edges with their colours, then no pair of colours appears twice. The smallest number of colours for it to exist is called the harmonious total chromatic number of $G$, denoted by $h_t(G)$. Here, we give a general upper bound for $h_t(G)$ in terms of the order $n$ of $G$. Our two main results are obvious consequences of the computation of the harmonious total chromatic number of the complete graph $K_n$ and of the complete multigraph $\lambda K_n$, where $\lambda$ is the number of edges joining each pair of vertices of $K_n$. In particular, Araujo-Pardo et al. have recently shown that $\frac{3}{2}n\leq h_t(K_n) \leq \frac{5}{3}n +\theta(1)$. In this paper, we prove that $h_t(K_{n})=\left\lceil \frac{3}{2}n \right\rceil$ except for $h_t(K_{1})=1$ and $h_t(K_{4})=7$; therefore, $h_t(G) \le \left\lceil \frac{3}{2}n \right\rceil$, for every graph $G$ on $n>4$ vertices. Finally, we extend such a result to the harmonious total chromatic number of the complete multigraph $\lambda K_n$ and as a consequence show that $h_t(\mathcal{G})\leq (\lambda-1)(2\left\lceil\frac{n}{2}\right\rceil-1)+\left\lceil\frac{3n}{2}\right\rceil$ for $n>4$, where $\mathcal{G}$ is a multigraph such that $\lambda$ is the maximum number of edges between any two vertices.Comment: 11 pages, 5 figure

Mechanisms involved in the promoting activity of fibroblasts in HTLV-1-mediated lymphomagenesis: Insights into the plasticity of lymphomatous cells

Among the mechanisms leading to progression to Adult T-cell Leukaemia/Lymphoma in Human T-cell Leukaemia Virus type 1 (HTLV-1)-infected subjects, the contribution of stromal components remains poorly understood. To dissect the role of fibroblasts in HTLV-1-mediated lymphomagenesis, transcriptome studies, cytofluorimetric and qRT-PCR analyses of surface and intracellular markers linked to plasticity and stemness in coculture, and in vivo experiments were performed. A transcriptomic comparison between a more lymphomagenic (C91/III) and the parental (C91/PL) cell line evidenced hyperactivation of the PI3K/Akt pathway, confirmed by phospho-ELISA and 2-DE and WB analyses. C91/III cells also showed higher expression of mesenchymal and stemness genes. Short-term coculture with human foreskin fibroblasts (HFF) induced these features in C91/PL cells, and significantly increased not only the cancer stem cells (CSCs)-supporting CD10+GPR77+ HFF subpopulation, but also the percentage of ALDH1bright C91/PL cells. A non-cytotoxic acetylsalicylic acid treatment decreased HFF-induced ALDH1bright C91/PL cells, downregulated mesenchymal and stemness genes in cocultured cells, and delayed lymphoma growth in immunosuppressed mice, thus hindering the supportive activity of HFF on CSCs. These data suggest that crosstalk with HFF significantly intensifies the aggressiveness and plasticity of C91/PL cells, leading to the enrichment in lymphoma-initiating cells. Additional research is needed to better characterize these preliminary findings

Comprehensive characterization of intraductal oncocytic papillary neoplasm (IOPN) of the pancreas: a systematic and critical review

Intraductal oncocytic papillary neoplasm (IOPN) of the pancreas is a recently recognized pancreatic tumor. Here, we aimed to determine the most essential features with the systematic review tool. PubMed, SCOPUS, and Embase were searched for studies reporting data on pancreatic IOPN. The clinicopathological, immunohistochemical, and molecular data were extracted and summarized. Then, a comparative analysis of the molecular alterations of IOPN with those of pancreatic ductal adenocarcinoma and intraductal papillary mucinous neoplasm from reference cohorts (including The Cancer Genome Atlas) was conducted. The key findings from 414 IOPNs were as follows: 1) Clinicopathological Features: Male-to-female ratio was 1,5:1. Pancreatic head was the most common site (131/237, 55.3%), but a diffuse tumor extension involving more than one pancreatic segment was described in about 1/5 of cases (49/237, 20.6%). The mean size was 45.5 mm. An associated invasive carcinoma was present in 50% of cases (168/336). In those cases, most tumors were pT1/pT2 and pN0 (>80%), and vascular invasion was uncommon (20.6%). Regarding survival, more than 90% of patients were alive after surgical resection. 2) Immunohistochemical and Molecular Features: The most expressed mucins were MUC5AC (110/112, 98.2%) and MUC6 (78/84, 92.8%). Compared with pancreatic ductal adenocarcinoma and intraductal papillary mucinous neoplasm, the classic pancreatic drivers KRAS, TP53, CDKN2A, SMAD4, and GNAS were less altered in IOPN (p<0.01). Moreover, fusions involving PRKACA or PRKACB genes were detected in all of 68 cases examined, with PRKACB::ATP1B1 as the most common (27/68 cases, 39.7%). These genomic events emerged as an entity-defining molecular alteration of IOPN (p<0.01). Thus, such fusions represent a promising biomarker for diagnostic purposes. Recent evidence also suggests their role in influencing the acquisition of oncocytic morphology. IOPN is a distinct pancreatic neoplasm with specific clinicopathological and molecular features. Considering the clinical/prognostic implications, its recognition is essential for pathologists and, ultimately, patients' management

Graphs with large palette index

Given an edge-coloring of a graph, the palette of a vertex is defined as the set of colors of the edges which are incident with it. We define the palette index of a graph as the minimum number of distinct palettes, taken over all edge-colorings, occurring among the vertices of the graph. Several results about the palette index of some specific classes of graphs are known. In this paper we propose a different approach that leads to new and more general results on the palette index. Our main theorem gives a sufficient condition for a graph to have palette index larger than its minimum degree. In the second part of the paper, by using such a result, we answer to two open problems on this topic. First, for every r odd, we construct a family of r-regular graphs with palette index reaching the maximum admissible value. After that, we construct the first known family of simple graphs whose palette index grows quadratically with respect to their maximum degree

Epithelial-mesenchymal transition in undifferentiated carcinoma of the pancreas with and without osteoclast-like giant cells

Undifferentiated carcinoma (UC) and undifferentiated carcinoma with osteoclast-like giant cells (UCOGC) are peculiar variants of pancreatic ductal adenocarcinoma (PDAC), characterized by hypercellularity and absence of glandular patterns. The inflammatory microenvironment is peculiar in UCOGC, since it is dominated by macrophages and osteoclast-like giant cells. However, from a molecular point of view, both UC and UCOGC are very similar to conventional PDAC, sharing alterations of the most common genetic drivers. Clinically, UC usually show a worse prognosis, whereas UCOGC may show a better prognosis if it is not associated with a PDAC component. To highlight potential biological differences between these entities, we investigated the role of the epithelial to mesenchymal transition (EMT) in UC and UCOGC. Specifically, we analyzed the immunohistochemical expression of three well-known EMT markers, namely Twist1, Snai2, and E-cadherin, in 16 cases of UCOGC and 10 cases of UC. We found that EMT is more frequently activated in UC (10/10 cases) than in UCOGC (8/16 cases; p = 0.05). Furthermore, in UCOGC, EMT was activated with a higher frequency in cases with an associated PDAC component. Snai2 was the most frequently and strongly expressed marker in both tumor types (10/10 UC, 8/16 UCOGC), and its expression was higher in UC than in UCOGC (mean immunohistochemical score: 4.8 in UC vs. 2.1 in UCOGC, p < 0.01). Our results shed new light on the biology of UC and UCOGC: EMT appeared as a more important process in UC, and Snai2 emerged as a central EMT effector in this setting

KRAS wild-type pancreatic ductal adenocarcinoma: molecular pathology and therapeutic opportunities

Pancreatic ductal adenocarcinoma (PDAC) is a deadly disease, whose main molecular trait is the MAPK pathway activation due to KRAS mutation, which is present in 90% of cases. The genetic landscape of KRAS wild type PDAC can be divided into three categories. The first is represented by tumors with an activated MAPK pathway due to BRAF mutation that occur in up to 4% of cases. The second includes tumors with microsatellite instability (MSI) due to defective DNA mismatch repair (dMMR), which occurs in about 2% of cases, also featuring a high tumor\ua0mutational burden. The third category is represented by tumors with kinase fusion genes, which marks about 4% of cases. While therapeutic molecular targeting of KRAS is an unresolved challenge, KRAS-wild type PDACs have potential options for tailored treatments, including BRAF antagonists and MAPK inhibitors for the first group, immunotherapy with anti-PD-1/PD-L1 agents for the MSI/dMMR group, and kinase inhibitors for the third group. This calls for a complementation of the histological diagnosis of PDAC with a routine determination of KRAS followed by a comprehensive molecular profiling of KRAS-negative cases