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A graph is called 1-planar if it can be drawn in the plane so that each its edge is crossed by at most one other edge. In the paper, we study the existence of subgraphs of bounded degrees in 1-planar graphs. It is shown that each 1-planar graph contains a vertex of degree at most 7; we also prove that each 3-connected 1-planar graph contains an edge with… (More)

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)

We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree ∆(G) ≤ D, D ≥ 12 is proved to be upper bounded by 6+ 2D+12 D−2 ((D−1) (t−1) −1). This improves a recent bound 6 + 3D+3 D−2 ((D − 1) t−1 − 1), D ≥ 8 by Jendrol' and… (More)