Abstract A 51-year-old man was diagnosed with renal cell carcinoma (RCC) of the left kidney seven years ago. The tumour staged as T2N0M0 and patient underwent an uncomplicated open radical nephrectomy. RCC classified clear he kept on biannual surveillance thereafter. Six after nephrectomy, had reported a 5-month history worsening pain globally restricted movement in their right shoulder. palpab...

It is known that for every integer k ≥ 4, each k-map graph with n vertices has at most kn − 2k edges. Previously, it was open whether this bound is tight or not. We show that this bound is tight for k = 4, 5. We also show that this bound is not tight for large enough k (namely, k ≥ 374); more precisely, we show that for every 0 < < 3 328 and for every integer k ≥ 140 41 , each k-map graph with ...

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