The saturation number for the length of degree monotone paths

نویسندگان

  • Yair Caro
  • Josef Lauri
  • Christina Zarb
چکیده

A degree monotone path in a graph G is a path P such that the sequence of degrees of the vertices in the order in which they appear on P is monotonic. The length (number of vertices) of the longest degree monotone path in G is denoted by mp(G). This parameter, inspired by the well-known ErdősSzekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter mp(G). We call G saturated if, for every edge e added to G, mp(G + e) > mp(G), and we define h(n, k) to be the least possible number of edges in a saturated graph G on n vertices with mp(G) < k, while mp(G+ e) ≥ k for every new edge e. We obtain linear lower and upper bounds for h(n, k), we determine exactly the values of h(n, k) for k = 3 and 4, and we present constructions of saturated graphs.

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عنوان ژورنال:
  • Discussiones Mathematicae Graph Theory

دوره 35  شماره 

صفحات  -

تاریخ انتشار 2015