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In: Advanced Math

Determine whether the following statement about graph theory is true or false. (1) If a graph...

Determine whether the following statement about graph theory is true or false.

(1) If a graph with m vertices is connected, then there must be at least m-1 edges.

(2) If a graph with m vertices has at least m−1 edges, then the graph must be connected.

(3) A simple undirected graph must contain a cycle, if it has m vertices with at least m edges.

(4) A graph must contain at least m edges, if it has m vertices and contains a cycle.

(5) The number of proper vertex colorings for any bipartite graph is at most two.

(6) The graph has an Euler tour, if all the vertices of a graph have an even degree.

(7) In a tournament graph, there always exists a directed Hamiltonian cycle.

(8) A simple undirected graph a Hamiltonian cycle, if it has an Euler tour.

(9) A simple undirected graph has an Euler tour, if it has a Hamiltonian cycle.

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Expert Solution

Colby Messinger answered 2 years ago
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