Lightweight paths in graphs

Let k be a positive integer, G be a graph on V(G) containing a path on k vertices, and w be a weight function assigning each vertex v ∈ V(G) a real weight w(y). Upper bounds on the weight [formula] of P are presented, where P is chosen among all paths of G on k vertices with smallest weight

On longest cycles in essentially 4-connected planar graphs

A planar 3-connected graph G is essentially 4-connected if, for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. Jackson and Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that . For a cubic essentially 4-connected planar graph G, Grünbaum with Malkevitch, and Zhang showed that G has a cycle on at least ¾ n vertices. In the present paper the result of Jackson and Wormald is improved. Moreover, new lower bounds on the length of a longest cycle of G are presented if G is an essentially 4-connected planar graph of maximum degree 4 or G is an essentially 4-connected maximal planar graph

Paths Of Low Weight In Planar Graphs

The existence of subgraphs of low degree sum of their vertices in planar graphs is investigated. Let K1;3 , a subgraph of a graph G, be an (x; a; b; c)-star, a star with a central vertex of degree x and three leaves of degrees a, b and c in G. The main results of the paper are: 1. Every planar graph G of minimum degree at least 3 contains an (x; a; b; c)-star with a b c and (i) x = 3, a 10, or (ii) x = 4, a = 4, 4 b 10, or (iii) x = 4, a = 5, 5 b 9, or (iv) x = 4, 6 a 7, 6 b 8, or (v) x = 5, 4 a 5, 5 b 6 and 5 c 7, or (vi) x = 5 and a = b = c = 6