Abstract :
It is known that under appropriate assumptions, each plane graph contains a vertex of degree at most 5 and a pair of adjacent vertices with degree sum at most 13. Two structural assumptions are established for a plane graph which together guarantee the existence of a triple of pairwise adjacent vertices with restricted degree sum. As shown by constructions, if any of these assumptions is violated, the degree sum of each three pairwise adjacent vertices may be arbitrarily large. As for a quadruple of pairwise adjacent vertices, it can hardly be forced in a plane graph by means of any reasonable restrictions.
