CompSci 163/265, Spring 2016, Homework 7

  1. Let $G$ be a graph wth six vertices (numbered 1, 2, 3, 4, 5, and 6) and nine edges: a cycle 1–2–3–4–5–6 and three diagonals 1–3, 3–6, and 4–6. Find a set of intervals for which this is the interval graph.

  2. (163 students:) For the same graph $G$ as the previous question, list the vertices in the order produced by a lexicographic breadth first search, starting from vertex 1.

    (265 students:) In class we went over an example of a bipartite graph (a crown graph) for which a greedy coloring with a bad vertex ordering can produce a non-optimal coloring. But the bad ordering that we used for this graph was not the lexicographic breadth first search ordering (which works well for interval graphs). Prove that, when a breadth first ordering is used (regardless of whether it is the lexicographic breadth first search ordering), the greedy coloring algorithm with this ordering will optimally color every bipartite graph.

  3. For each of the five Platonic solids (the tetrahedron, cube, regular octahedron, regular dodecahedron, and regular icosahedron) state whether the graph of its vertices and edges is (a) chordal, (b) bipartite, (c) perfect but neither chordal nor bipartite, (d) none of the above.

  4. Find an example of a chordal graph, and a breadth first search ordering of the graph, such that some vertex of the graph has a set of earlier neighbors that do not form a clique. (This can't happen for a lexicographic BFS ordering so your ordering will need to be non-lexicographic.)