Prove that every simple, triangle-free, planar graph is 4-colorable. (Hint: show first that such a graph has a vertex of degree at most 3.)

18. Show that in a simple graph with at least two vertices there must be two vertices that have the same degree.

5. Another Proof Consider an undirected, unweighted, simple graph G. Suppose the longest simple path in G has length 10 and goes from node u to node v. Prove that the degree of node u is at most 10.

Is there a connected planar graph with an odd number of faces where every vertex has degree 6? Prove your answer.

Prove or disprove that a graph is bipartite if and only if every induced cycle has even length.

A graph is biconnected if every pair of vertices is connectedby two disjoint paths. An articulation point in a connected graph is a vertex that woulddisconnect the graph if it (and its adjacent edges) were removed. Prove that any graphwith no articulation points is biconnected. Hint : Given a pair of vertices s and t and apath connecting them, use the fact that none of the vertices on the path are articulationpoints to construct two disjoint paths connecting s and t.

Find reflexive, symmetric and transitive closure graphs for following graphs

Discrete math Draw 2 isomorphic graphs with graph order of degree 3 which has 8 vertices.

Let H be a planar graph with n vertices and m edges, where n ≥ 5. Prove that if H does not have any cycles of length 3 and 4 then 3m ≤ 5n - 10.

Please solve it in your own way. Do not copy chegg or others methods /