Word Count : 1111

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Let’s go step-by-step. --- ## (a) Verify Euler’s formula for the following plane graph. *(No figure provided in the text, but I’ll assume a standard example of a plane graph.)* Euler’s formula for a connected planar graph is \[ n - m + f = 2 \] where \(n\) = number of vertices, \(m\) = edges, \(f\) = faces. If a specific graph is given, you’d count: - vertices \(n\) - edges \(m\) - faces \(f\) (including the outer face) and check \(n - m + f = 2\). Since no diagram is given here, I can’t compute specific numbers, but that’s the verification method. --- ## (b) Check whether the line graph of \(C_5 \times K_2\) is planar or not. First: \(C_5\) is a 5-cycle. \(C_5 \times K_2\) is the Cartesian product of \(C_5\) and \(K_2\). This graph is a 3-regular graph with \(n = 5 \times 2 = 10\) vertices, known as the prism over \(C_5\) (pentagonal prism). Step 1: Degree of vertices in \(C_5 \times K_2\): Each vertex: degree \( \deg_{C_5}(v) + \deg_{K_2}(w) = 2 + 1 = 3\). Yes, 3-regular, 10 vertices. Number of edges \(m = \frac{10 \times 3}{2} = 15\). Step 2: Line graph \(L(G)\) of \(G = C_5 \times K_2\): Vertices of \(L(G)\) correspond to edges of \(G\) → \(m = 15\) vertices in \(L(G)\). _____ ______ ____ ________ ____ ________.
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