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Graph Coloring Problem

Please use words to describe the solution process: Let G be a graph with n vertices that is not a complete graph. Prove that x (G) < n HINT: If G does not contain k3 as a subgraph, then every face must have degree at least 4. *(Please see attachment for proper symbols)

Subject:

Math

Topic:

Discrete Structures

Posting ID:

29709

OTA ID:

104455

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Graph Coloring Problem

Please use words to describe the solution process: Let G be a graph with exactly one cycle. Prove that x(G) is less than or equal t0 3. *(Please see attachment for proper symbols)

Subject:

Math

Topic:

Discrete Structures

Posting ID:

29710

OTA ID:

104455

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Propositional Logic : DeMorgan's Laws and Truth Tables

Please see the attached file for the fully formatted problems. Verify DeMorgan's laws (equation 1 and 2 below) using truth tables. Prove the generalized DeMorgan’s laws: (1) (NOT(p1 p2 .... pk)) (2) (NOT(p1+p2+...+pk)) by induction on k, using the basic laws: NOT(pq) NOT(p+q) Then, justify the generalized laws informally by describing what 2k row truth tables for each expression and their subexpressions look like.

Subject:

Math

Topic:

Discrete Structures

Posting ID:

29903

OTA ID:

101298

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Propositional Logic : Commutitivity of Sums and Products

Please see the attached file for the fully formatted problems. Prove that: p1+p2+...+pn is equivalent to the sum (logical OR) of the pi’s in any order. and p1p2 ... pn is equivalent to the product (logical AND) of the pi’s in any order.

Subject:

Math

Topic:

Discrete Structures

Posting ID:

29904

OTA ID:

104632

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Discrete Structures : Coloring

Let G be a properly colored graph and let us suppose that one of the colours used is red. The set of all red-coloured vertices have a special property. What is it? Graph colouring can be thought of as partitioning V(G) into subsets with this special property. (See attachment for full background)

Subject:

Math

Topic:

Discrete Structures

Posting ID:

30125

OTA ID:

103300

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