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A subgraph of a plane graph is light if each of its vertices has a small degree in the entire graph. Consider the class ~¢-(5) of plane triangulations of minimum degree 5. It is known that each G C,Y-(5) contains a light triangle. From a recent result of Jendrol' and Madaras the existence of light cycles C4 and C5 in each G C.~(5) follows. We prove here(More)
A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is well known that a plane graph of minimum degree five contains light edges and light triangles. In this paper we show that every plane graph of minimum degree five contains also light stars K 1,3 and K 1,4 and a light 4-path P 4. The(More)
A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains (1) a 5-vertex adjacent to an ≤ 6-vertex, (2) a 4-cycle whose every vertex has degree at most 9, (3) a K 1,4 with all vertices having degree at most 11.
The weight of a path in a graph is defined to be the sum of degrees of its vertices in entire graph. It is proved that each plane triangulation of minimum degree 5 contains a path P 5 on 5 vertices of weight at most 29, the bound being precise, and each plane triangulation of minimum degree 4 contains a path P 4 on 4 vertices of weight at most 31.(More)