A graph is called supermagic if there is a labeling of edges where the edges are labeled with consecutive distinct positive integers such that the sum of the labels of all edges incident with any vertex is constant. A graph G is called degree-magic if there is a labeling of the edges by integers 1, 2, ..., |E(G)| such that the sum of the labels of the edges incident with any vertex v is equal t...

A classical theorem by Erdős, Kleitman and Rothschild on the structure of triangle-free graphs states that with high probability such graphs are bipartite. Our first main result refines this theorem by investigating the structure of the ’few’ triangle-free graphs which are not bipartite. We prove that with high probability these graphs are bipartite up to a few vertices. Similar results hold if...

In this paper it is shown that a class of almost-median graphs that includes all planar almost-median graphs can be recognized in O(m log n) time, where n denotes the number of vertices and m the number of edges. Moreover, planar almost-median graphs can be recognized in linear time. As a key auxiliary result we prove that all bipartite outerplanar graphs are isometric subgraphs of the hypercub...