Posted: June 30th, 2022

TCSS 321 Discrete Mathematics

Question:

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A

N(S)

[Satisfies Hall’s condition.]

Use the generalized version Menger’s theory (see Theorem 2.9) to prove Hall’s argument.

Let’s say that Hall’s condition is no longer applicable. Instead, we have the following condition to some positive integer k.

N(S)

(1) Display that G has a collection stars on k+ 1 vertices, which saturate A.

A star on k+1 is a graph that has k vertices at degree 1 joined to one vertex of degree 1.

If G is an nvertex graph with maximum degree, then prove it.

If G is a n-vertex graph with maximum degree?, then

Consider a 5-regular (i.e., graphs where all vertices have degree 5), which has no 1-factor.

(Graduate exercise). Use Tutte’s theorem for Hall’s proof.

Answer:

Make sure to prepare your answers using LaTeX.

For help with preparing your HW file, refer to hwtemplate.tex

Please create a page for each solution.

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A

Let us suppose that we have a set for each set S A.

A

A

Add d new vertex to B, each one connected to all vertex in A.

Let G0 be your new graph.

NG0(S)

A

V (G)

Use the generalized version Menger’s theory (see Theorem 2.9) to prove Hall’s argument.

N(S)

G can be a bipartite graph, with vertex classes X or Y.

Two new vertices are added to G: a and b. We join a to all elements in X and b together to all elements in Y.

Let G0 be this graph.

Let C be the set of vertices that separate a and b in G0.

C

N(X C)

X

Let’s say that Hall’s condition is no longer applicable. Instead, we have the following condition to some positive integer k.

N(S)

Show that G has a set of stars on k+ 1 vertices, which saturate A.

A star on k+1 is a graph that has k vertices at degree 1 joined to one vertex at degree k.

Let G be a bipartite diagram with bipartition (V1,V2), and let M be the maximum matching of G (Hall’s condition).

Denoting U is the set M of unsaturated verticles in V1, and Z denoting the set all vertices connected via M-alternating paths to U.

If S=Z/V1 and T=Z/V2, then, as per the half theorem we know that each vertex in T has M-saturated and?

(s) =T, thus G contains a collection stars on k+ 1 vertices which saturate. A star on k+ 1 is a graph that has k vertices all joined to one vertex of degree 1.

If G is an nvertex graph with maximum degree, then prove it.

If G is a n-vertex graph with maximum degree?, then

G is the n-vertex graph of maximum degree.

If G is a graph with no vertex of degree 0, then the upper bound will be immediately and clearly sharp.

To verify the lower bound, induction is used to determine the size of the connected graph. If m >= 2, then the lower bound follows.

Assuming that the lower bound applies to all connected graphs with positive sizes not exceeding 2 and G being a connected graph of order 1 having a size of k+1,

If G has a edge of e on its cycle, then

b1 (G) >= G-e, otherwise G is a tree.

If G=K1, then n-1, then G contains

If GK1, n-1 is true, then G has an edge e such (G-e), which contains two nontrivial parts G1 and G2.

Let ni denote Gi’s order, i=1, 2, and then apply the induction hypothesis for G1 and G2.

This means that b1 (G) >= G1 (G1) >= B1(G2) =

Consider a 5-regular (i.e., graphs where all vertices have degree 5), which has no 1-factor.

Solution

Hall Theorem: A bipartite diagram G with partition (A and B) has a matching value of

A =S A.?N(S).

T

X

Assuming H has a 1 factor (i.e.

Let H be a perfect matching and M be the edges of this matching that are incident with vertices on X.

M

The edges of M match every vertex in X with some vertex at Y, so the edges of H are not changed.

Although there might be vertices not matching M’s Y, the construction of H ensured that the graph induced from these vertices was a clique that is on an even number vertices. This allows us to complete the matching.

Assuming that G fulfills Hall’s condition, let us assume that T V (H) is true.

X

Y

Consider that Y 6T is a clique in H. Since Y has all the vertices of Y T, there is one component B to H T. Let S = (B).

S

V (B)

S

We have done it again.

N(S)

X

Refer to

5-regular graphs can be 3-colored with positive probabilities.

Algorithms-ESA 2008 (Brodal, Leonardi, Eds.

Random Graphs: Wiley. New York.

M (2010).

For the coloring problem on random graphs, Phys., threshold values, stability analysis, and high-q asymptotic are all available.

Rev.

Random K-Satisfiability – From an Analytic Solution, to a new efficient algorithm. Phys.

Rev.