Question

(d) graph $k_{2,3}$ (complete bipartite graph)
☐ regular
☐ complete bipartite
☐ complete
☐ bipartite
☐ has a cycle
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Explanation:

• Regular: A regular graph has all vertices with the same degree. In \(K_{2,3}\), vertices in the set of size 2 have degree 3, and vertices in the set of size 3 have degree 2. So not regular.
• Complete bipartite: By definition, \(K_{m,n}\) is a complete bipartite graph, so \(K_{2,3}\) is complete bipartite.
• Complete: A complete graph has every pair of distinct vertices adjacent, but \(K_{2,3}\) is bipartite (edges only between the two sets), not a complete graph (which is not bipartite for \(n\geq3\)).
• Bipartite: Complete bipartite graphs are bipartite (they can be divided into two disjoint sets with no edges within a set), so \(K_{2,3}\) is bipartite.
• Has a cycle: A bipartite graph has a cycle if and only if it has an even - length cycle. \(K_{2,3}\) has cycles (e.g., take two vertices from the set of size 2 and two vertices from the set of size 3, we can form a cycle of length 4, which is even). Also, for a bipartite graph \(K_{m,n}\) with \(m\geq2\) and \(n\geq2\), it has cycles.

Answer:

• Complete bipartite (checked)
• Bipartite (checked)
• Has a cycle (checked)