Determine whether the following statement about graph theory istrue or false.
(1) If a graph with m vertices is connected, then there must beat least m-1 edges.
(2) If a graph with m vertices has at least m?1 edges, then thegraph must be connected.
(3) A simple undirected graph must contain a cycle, if it has mvertices with at least m edges.
(4) A graph must contain at least m edges, if it has m verticesand contains a cycle.
(5) The number of proper vertex colorings for any bipartitegraph is at most two.
(6) The graph has an Euler tour, if all the vertices of a graphhave an even degree.
(7) In a tournament graph, there always exists a directedHamiltonian cycle.
(8) A simple undirected graph a Hamiltonian cycle, if it has anEuler tour.
(9) A simple undirected graph has an Euler tour, if it has aHamiltonian cycle.
