MATH 443
Problems #7
Due Thursday Nov 21.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
35. Let L(G) denote the line graph of G. Let G be a simple graph with χ(L(G)) = 2. Show that
G is a vertex disjoint union of paths and even cycles.
36. Let G be a simple graph on n vertices. Let Gc denote the complement of G; namely V (Gc ) =
V (G), E(Gc ) = E(Kn )\E(G). Show that
χ(G) + χ(Gc ) ≤ n + 1
(try induction on n)
37. A hamiltonian cycle in a graph is a spanning cycle. Show that the complete tripartite graph
Kr,s,t (with r ≤ s ≤ t) has a hamiltonian cycle if and only if t ≤ r + s.
38. Let G be 2-connected. A plane graph is 2-face colourable if and only if G is eulerian.
39. Consider a 2-connected plane graph G for which all faces have size 3. Show that G is eulerian
if and only if χ(G) = 3.
40. Consider a plane graph G which has a spanning cycle (a hamiltonian cycle). Show that G has
a 4-face colouring without using the 4-colour theorem.
41. Let G be a graph with degree sequence d1 , d2 . . . . , dn where d1 ≥ d2 ≥ · · · ≥ dn and n = |V (G)|.
Then
χ(G) ≤ max min{di + 1, i}.
i: 1≤i≤n
42. Let Wn be the wheel graph on n vertices. Show that χ(Wn ; k) = k(k − 2)n−1 + (−1)n−1 k(k − 2).