Every planar graph with nine points has a nonplanar complement

Every rayless graph has an unfriendly partition

We prove that every rayless graph has an unfriendly partition.

Planar Graphs on Nonplanar Surfaces

It is shown that embeddings of planar graphs in arbitrary surfaces other than the 2-sphere have a special structure. It turns out that these embeddings can be described in terms of noncontractible curves in the surface, meeting the graph in at most two points (which may taken to be vertices of the graph). The close connection between the homology group of the surface and the planar graph embedd...

Every Body Has a Brain

[1] http://ebhb.morphonix.com/ [2] http://fireflyfoundation.wordpress.com/brain-games/every-body-has-a-brain/ [3] http://dev.cdgr.ucsb.edu/organizations/morphonix [4] http://dev.cdgr.ucsb.edu/category/genre/educational [5] http://dev.cdgr.ucsb.edu/category/genre/music/dance [6] http://dev.cdgr.ucsb.edu/category/topics/anatomy [7] http://dev.cdgr.ucsb.edu/category/topics/brain [8] http://dev.cdg...

Every Algorithm Has a POV

Fixed points for Chatterjea contractions on a metric space with a graph

In this work‎, ‎we formulate Chatterjea contractions using graphs in metric spaces endowed with a graph ‎‎and‏ ‎‎‎‎investigate ‎the ‎existence‎ ‎of ‎fixed ‎points ‎for such mappings ‎under two different hypotheses‎. We also discuss the uniqueness of the fixed point. The given result is a generalization of Chatterjea's fixed point theorem from metric spaces to metric spaces endowed with a graph.