a few problems on graph theory
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8/13/2019 A Few Problems on Graph Theory.
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Problems Set 5: Graph Theory
1 Graph Theory
1. Show that every connected graph has at least two verticesu and v such thatGuand Gv
are connected.
2. Show that for any graphG = (V, E) the vertex setVcan be partitioned into two sets V1 and
V2 such that
e(V1) +e(V2)
|E|
2
wheree(Vi) means the number of edges in Ewith both end points in Vi.
3. Prove that a regular bipartite graph of degree at least 2 does not contain a bridge.
4. Let G be a graph with minimum degree 2. Show that there exist a connected graph with
same degree sequence.
5. Let T1, . . . , T k be subtrees of a tree Tsuch that for all i, j the trees Ti and Tj have a vertex
in common. Show that Thas a vertex that is in all Ti.
6. LetG be a planar graph, with edges colored redand blue. Show that there is a vertex v such
that going round the vertex in a clockwise direction we encountered no more than two change
of colors.
7. IfG = (V, E) is a graph on nvertices such that all the vertices have even degree. Show that
the edge set Ecan be partitioned into pairwise disjoint sets C1, C2, . . . , C k such that for all
1 i k the subgraphs (V, Ci) is a cycle and a collection of isolated vertices.
8. If a graph has maximum degree less than or equal to k then it is (k+ 1)-colorable.
9. A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in
an undirected complete graph. That is, it is a directed graph in which every pair of vertices
is connected by a single directed edge. Prove that a tournament always has a Hamiltonianpath.
10. If in a directed graph number of incoming edges is equal to number of outgoing edges then
the graph has an Eulerian Path.
11. Prove that then-dimensional cube graph has a Hamiltonian path.
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12. Prove that 101 999 grid does not have a Hamiltonian Cycle.
13. Prove that a graph with n vertices and n+ 2 edges in planar.
14. How many 4-cycles are in Kn,n?
15. Prove that eitherG or G is connected.
16. A chord of a cycle is an edge that connects two non-adjacent vertices in the cycle. Prove that
if every node ofG has degree 3 then Gcontains a cycle with a chord.
17. IfG is a bipartite graph with m nodes on each side. If each node has degree more than m/2
then prove that it has a perfect matching.
18. Prove that a planar bipartite graph has at most 2n 4 edges.
19. If every node has degree dand at least one vertex has degree strictly less than d then prove
that the graph is d-colorable.
20. If every face of a planar graph has even number of edges then prove that the graph is bipartite.
21. Prove that there is a tournamentT withn players and at leastn!2(n1) Hamiltonian paths.
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