4. NEZ is the mmber of vertices that have odd degree, then N must be an even number. Let G = (V, E) be a graph. Prove the Handshaking Theorem, namely that if

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.

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.)

Please help me solve this problem

Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.

Write down a vertex sequence for an Euler circuit or Euler path in the following graph, if either exists. If neither exist, state why that is so. a

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

5. Let G = (V,E) be a loop-free connected unicyclic graph (contains one single cycle).If G has five vertices of degree 2, two vertices of degree 3, and three vertices of degree 4, howmany pendant vertices (leaves) does it have. (Assuming that maximum degree in G is 4).

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.