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Prove that if deg(v) ≤ 4 for all vertices in an (undirected) graph G = (V,...

Prove that if deg(v) ≤ 4 for all vertices in an (undirected) graph G = (V, E), then we can orient all edges in E such that the in-degree of every vertex is at most 2.

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Proof : We already know that the sum of in-degree and out-degree of all the vertices in a directed graph is equal to the sum of degrees of all the vertices in its undirected version with equal number of edges.So, for an udirected graph with V vertices,with at most 4 degrees of each vertex, we have,

Sum of degrees ≤ 4*V ...(1)

But we also know that ,

Sum of In-degree (in directed graph) + Sum of Out-degree (in directed graph) = Sum of degrees (in undirected graph) ...(2)

Also, for a particular case in a graph we CAN have,

Sum of In-degree ≤ Sum of Out-degree ...(3)

So, for that scenario Using (2) & (3) we get,

2*Sum of In-degree (in directed graph) ≤ Sum of degrees (in undirected graph) ....(4)

Using (1) and (4) we get,

2*Sum of In-degree (in directed graph) ≤ 4*V   ...(5)

Or, using (5) we can write,

Sum of In-degree (in directed graph) ≤ 2*V

( Hence Proved ) That there exists a case in which we can orient the directed edges such that in-degree of every vertex is at most 2.

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