Question

Graph Theory

Answer #1

Let G be a graph or order n with independence number α(G) =
2.
(a) Prove that if G is disconnected, then G contains K⌈ n/2 ⌉ as
a subgraph.
(b) Prove that if G is connected, then G contains a path (u, v,
w) such that uw /∈ E(G) and every vertex in G − {u, v, w} is
adjacent to either u or w (or both).

Let G be a graph with vertex set V. Define a
relation R from V to itself as follows: vertex
u has this relation R with vertex v,
u R v, if there is a path in G from u to
v. Prove that this relation is an equivalence relation.
Write your proof with complete sentences line by line in a logical
order. If you can, you may write your answer to this
question directly in the space provided.Your presentation
counts.

Let G be an n-vertex graph with n ≥ 2 and δ(G) ≥ (n-1)/2. Prove
that G is connected and that the diameter of G is at most two.

let G be a connected graph such that the graph formed by
removing vertex x from G is disconnected for all but exactly 2
vertices of G. Prove that G must be a path.

Let G=(V,E) be a connected graph with |V|≥2
Prove that ∀e∈E the graph G∖e=(V,E∖{e}) is disconnected, then G
is a tree.

Prove or disapprove each of the following:
(a) Every disconnected graph has an isolated vertex.
(b) A graph is connected if and only if some vertex is connected
to all other vertices.
(c) If G is a simple, connected, Eulerian graph, with edges e, f
that are incident to a common vertex, then G has an Eulerian
circuit in which e and f appear consequently.

Graph Theory
Prove that if G is a graph with x(G-v-w)=x(G)-2 for every pair
of vertices v and w in G, then G is complete.
Hint: assume G is not complete.

Let G be a graph where every vertex has odd degree, and G has a
perfect matching. Prove that if M is a perfect matching of G, then
every bridge of G is in M.
The Proof for this question already on Chegg is wrong

Let G be a graph whose vertex set is a set V = {p1,
p2, p3, . . . , p6} of six people.
Prove that there exist three people who are all friends with each
other, or three people none of whom are friends with each
other.

GRAPH THEORY:
Let G be a graph which can be decomposed into Hamilton
cycles.
Prove that G must be k-regular, and that k must be even.
Prove that if G has an even number of vertices, then the edge
chromatic number of G is Δ(G)=k.

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