Practice 2012 III

7/28/2019 Practice 2012 III

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DEPARTMENT OF MATHEMATICS, IIT Guwahati

MA501: Discrete Mathematics, July - November 2012

Practice Problem Set III

1. Let G be a simple bipartite graph of n vertices and m edges. Show that m ≤ n2

4.

2. Show that

(a) every induced subgraph of a complete graph is complete;

(b) every subgraph of a bipartite graph is bipartite.

3. In a group of people, each girl knows exactly k boys (k > 0) and each boy knows exactly

k girls. Find the number of girls and the number of boys in the group.

4. Show that any two longest paths in a connected graph have a vertex in common.

5. Prove that a connected graph G remains connected after removing an edge e from G, if and only if e is in some cycle of G.

6. If the intersection of two paths in a graph is a disconnected subgraph, then show that

their union will contain at least one cycle.

7. Show that a graph G is disconnected then Gc is connected.

Definition: Let G be a graph. The line graph of G, denoted L(G), is the graph whose

vertex set is E (G) and for e, f ∈ E (G), ef ∈ E (L(G)) iff e and f are adjacent in G.

8. Show that the number of vertices and edges of the line graph L(G) of a simple graph G

are |E (G)| and

v∈V (G)

dG(v)

2

, respectively.

9. Let {x1, x2, . . . , xn} be a set of points in the plane such that the distance between any

two points is at least one. Show that there are at most 3n pairs of points at distance

exactly one.

10. Let G be a simple graph with n vertices and m edges such that m >(n−1)(n−2)

2. Prove

thatG

is connected. For eachn >

1, find a disconnected simple graphG

withn

verticesand m edges such that m = (n−1)(n−2)

2.

11. Prove that any two simple connected graphs with n vertices, all of degree two, are iso-

morphic.

12. Check that the two graphs in Figure 1 are isomorphic:

13. A graph G is called self-complementary if G and GC are isomorphic. Check if the graphs

P 4, P 5 and C 5 are self-complementary. Find a self-complementary graph on 5 vertices

(other than C 5). Show that if a graph G on n vertices is self-complementary then n ≡0, 1( mod 4).

14. Let n ∈ N be such that n ≡ 0, 1(mod 4). Draw a self-complementary graph on n vertices.



Figure 1: Two non-isomorphic graphs

15. Let G and H be two graphs, v ∈ V (G) and f : V (G) → V (H ) be an isomorphism. Show

that G − v and H − f (v) are also isomorphic.

16. Show that every simple graph on n vertices is isomorphic to a subgraph of K n.

17. An acyclic graph is called a forest. Show that

(a) each component of a forest is a tree;

(b) G is a forest with n vertices, m edges and ω components iff m = n − ω.

18. Show that if G is a forest with exactly 2k vertices of odd degree then there are k edge-

disjoint paths P 1, P 2 . . . , P k in G such that E (G) = E (P 1) ∪ E (P 2) ∪ . . . ∪ E (P k).

19. A saturated hydrocarbon is a molecule C mH n in which every carbon atom has four bonds,

every hydrogen atom has one bond, and no sequence of bonds forms a cycle. Show that,

for every positive integer m, C mH n can exist only if n = 2m + 2.

20. Let G be a connected graph. Prove that a maximal acyclic subgraph of G is a spanning

tree of G.

21. Prove that every n vertex graph with m edges has at least m − n + 1 cycles.

22. Let T be a tree. Show that T is bipartite and find a bipartition of T . Further, prove that

T has a pendent vertex in its larger partite set.

23. Let T be a tree. Prove that the vertices of T all have odd degree iff for all e ∈ E (T ),

both the components of T − e have odd order.

24. Prove that every connected subgraph of a tree is the induced one.

25. Let T be a tree. Prove that for every path P starting at a vertex v, there exists a pendent

vertex w such that P is a section of the (v, w)-path.

26. Prove or disprove: There is an Eulerian graph with even number of vertices and odd

number of edges.

27. Show that if G has no vertices of odd degree, then there are edge-disjoint cycles C 1, C 2, . . . , C m

such that E (G) = E (C 1) ∪ E (C 2) ∪ . . . ∪ E (C m).

28. Prove or disprove: Every Eulerian simple graph with an even number of vertices has an

even number of edges.



29. Prove or disprove: If G is an Eulerian graph with edges e, f that share a vertex, then G

has no Eulerian cycle in which e, f appear consecutively.

30. Prove that an Eulerian graph contains a unique Eulerian cycle iff the graph itself is a

cycle.

31. Show that if G is bi-partite with bi-partition X ∪ Y , where |X | = |Y |, then G is not

Hamiltonian.

32. A mouse eats his way through a 3 × 3 × 3 cube of cheese by tunneling through all of the

27 1 × 1 × 1 sub cubes. If he starts at one corner and always moves on to an uneaten

sub cube, can he finish at the centre of the cube?

33. For which values of r is K r,r Hamiltonian?

34. For n > 1, prove that K n,n has n!(n−1)!2

Hamiltonian cycles.

35. Let G be a simple graph on n vertices and Y = A + A2

+ A3

+ . . . + An−1

, where A is theadjacency matrix of G. Show that G is disconnected if and only if there is at least one

zero entry in Y .

36. Let A be the adjacency matrix of a graph G having e edges and t triangles. Then show

that

(a) trace(A) = 0, (b) trace(A2) = 2e, (c) trace(A3) = 6t.

37. Let A and B be the adjacency and incidence matrices of a graph G. Show that the

diagonal entries of both of A2

andBB t

are degrees of the vertices of G

.

38. Let A(G) and A(H ) are the adjacency matrices of the graphs G and H , respectively. Let

Let B(G) and B(H ) are the incidence matrices of the graphs G and H , respectively. If

G ∼= H then prove that

(a) A(G) is obtained from A(H ) by some permutations of rows and columns.

(b) B(G) is obtained from B(H ) by some permutations of rows and columns.

39. Let B be the incidence matrix of a graph G and let L be the adjacency matrix of its line

graph. Show that BtB = 2I + L, where I is an identity matrix of appropriate size.

40. Show that if any two odd cycles of G have a vertex in common then χ(G) ≤ 5.

41. Show that χ(G + H ) = χ(G) + χ(H ).

42. Prove that in some minimal proper coloring of a graph G, the neighborhood N (v) of every

vertex v is colored in the same color iff G is bipartite.

43. Prove that for any two different colors i, j of any minimal proper coloring of the vertices

of a graph G there exists an edge uv such that u has color i and v has color j.

44. Prove that in every proper coloring of the line graph L(G) of a graph G, every vertex is

adjacent to at most two vertices of the same color.



45. Prove that the number of edges of a graph G is at least χ(G)(χ(G)−1)2

.

46. Is it true that if deg(v) < χ(G) − 1 for some vertex v of a graph G then χ(G) = χ(G − v)?

47. Prove that χ(G) ≤ l + 1, where l is the length of a longest path of G.

48. Exhibit a graph G with a vertex v so that χ(G − v) < χ(G) and χ(G − v) < χ(G).

49. Let G be a connected planar graph with n vertices and m edges. Let k(≥ 3) be the lengthof the shortest cycle. Prove that m ≤ k(n−2)

k−2.

50. (a) Show that if G is a simple planar graph with |V (G)| ≥ 11, then Gc is not planar.

(b) Find a simple planar graph G with |V (G)| = 8 such that Gc is also planar.

51. Construct plane embedding of the graphs (a) K 5 − e, (b) K 3,3 − e such that the edges

are presented by straight line segments.

52. Prove that m ≤ 2( p + q − 2) for a planar bipartite m-edge graph with the sizes p, q (≥ 2)of the partite sets.

53. Prove that |E (H n)| ≤ 2n+1 − 4, where H n is a planar spanning subgraph of the n-cube

Qn, n ≥ 2.

54. For which r there exists a planar r-regular graph?

55. Prove that a planar connected cubic graph whose every face has at least 5 vertices is of

order at least 20.

56. Let G be a planar connected cubic graph. Prove that

i≥3

(6 − i)f i = 12, where f i is the

number of faces of G each of which is bounded i edges. [Here an edge that belongs to a

single face is counted twice in f i.]

Also prove that G has at least one face with at most 5 edges.

57. Prove that a planar graph with n vertices (n ≥ 4) has at least four vertices with degree

five or less.

58. Prove or disprove:

(a) Every subgraph of a planar graph is planar.

(b) Every subgraph of a non-planar graph is non-planar.

59. Every planar graph with fewer than 12 vertices has a vertex of degree at most 4.

60. Prove or disprove: There is no simple bipartite planar graph with minimum degree at

least 4.

Practice 2012 III

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