Question

graph theory

Prove that a graph of minimum degree at least k ≥ 2 containing no triangles contains a cycle of length at least 2k.

Answer #1

Let δ ≥ 2. Prove that every simple graph G satisfying δmin(G) ≥
δ and
containing no triangles contains a cycle of length at least
2δ.
Prove that this result is sharp by showing that we cannot guarantee
the existence of
a cycle of length at least 2δ + 1. Give a counterexample for each
δ.

Prove that for each k ≥ 1, a graph which is regular
with degree 2k can never have a bridge.

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.

1. Prove that {2k+1: k ∈ Z}={2k+3 : k ∈ Z}
2. Prove/disprove: if p and q are prime numbers and p < q,
then 2p + q^2 is odd (Hint: all prime numbers greater than 2 are
odd)

A K-regular graph G is a graph such that deg(v) = K for all
vertices v in G. For example, c_9 is a 2-regular graph, because
every vertex has degree 2. For some K greater than or equal to 2,
neatly draw a simple K-regular graph that has a bridge. If it is
impossible, prove why.

a
graph is regular of degree k if every vertex has the same degree,
k. show that G has a hamiltonian circuit if G has 13 vertices and
is regular of degree 6.

Graph Theory: What is the maximum degree a vertex can have in a
tree graph with 3 leaves.

(a) Let L be a minimum edge-cut in a connected graph G with at
least two vertices. Prove that G − L has exactly two
components.
(b) Let G an eulerian graph. Prove that λ(G) is even.

A Hamiltonian walk in a connected graph G is a closed spanning
walk of minimum length in G. PROVE that every connected graph G of
size m contains a Hamiltonian walk of length at most 2m in which
each edge of G appears at most twice.

(a) Prove that there does not exist a graph with 5 vertices with
degree equal to 4,4,4,4,2.
(b) Prove that there exists a graph with 2n vertices with
degrees 1,1,2,2,3,3,..., n-1,n-1,n,n.

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