(Solved): graph theory (a) Draw the complete bipartite graph \( K_{1,3} \), the cycle \( C_{4} \), and the Car ...

(a) Draw the complete bipartite graph \( K_{1,3} \), the cycle \( C_{4} \), and the Cartesian product \( K_{1,3} \square C_{4} \). Show each is bipartite by providing a proper 2-coloring of each graph. (b) By examining the colorings in part (a), make a conjecture describing a pattern for how to determine the color of a vertex \( x_{i} \) in the Cartesian product based on the color of vertex \( x \) in the "letter graph" and the color of vertex \( i \) in the "number graph". (c) Draw the path graphs \( P_{3} \) and \( P_{4} \) and the Cartesian product \( P_{3} \square P_{4} \). Properly 2-color the \( P_{3} \) and \( P_{4} \), and then use your conjecture from part (b) to color the vertices of \( P_{3} \square P_{4} \). Verify the coloring is proper, or modify your conjecture as needed.