## Graph TheoryTrouble in the Barker's Class |

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adjacent edges blue edges blue triangle China Normal University color common edge common vertex complete the proof connected graph contains a Hamiltonian contains a monochromatic convex polygon corresponding vertices cycle whose length deleting East China Normal edges joining edges of G Euler's Formula Example exists faces G contains G is connected graph contains graph G graph must contain graph theory Hamiltonian chain Hamiltonian cycle Hamiltonian graph incident induction International Mathematical Olympiad irrational numbers join the corresponding least lifting one's pen line segments Mathematical Olympiad monochromatic triangle number of edges number of vertices obtain a graph odd vertices outdegree pendant vertices persons Pigeonhole Principle planar graph plane players polyhedron problem Prove red edges red triangle shakes hands simple graph subdivision graph subset teams three edges three vertices tree Turan's Theorem two-color complete graph vertex set vertex whose degree vertices adjacent vertices of G