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Chapter 11: Problem 2

Characterize the type of graph in which an Euler trail (circuit) is also a Hamilton path (cycle).

Short Answer

Expert verified
The type of graph where an Euler trail or circuit could also serve as a Hamilton path or cycle is a simple graph where each node has a minimum degree of two and can form a Hamilton cycle.

Step by step solution

01

Understanding Euler Circuit and Hamiltonian Cycle

The fundamental rule for Euler circuits is that a graph \( G \) has an Euler circuit if and only if \( G \) is connected and each node has an even degree. A Hamiltonian cycle is a cycle in \( G \) that visits every node once with no repetition. Therefore, apart from isolated nodes, to make a cycle, the minimum degree of each node should be 2.
02

Comparing and Contrasting

Comparing these two, we observe that both need a minimum node degree of two, which indicates a cycle. In a simple graph, if each node is of minimum degree two, we get a cycle. When a Hamiltonian cycle is present in a graph, it implies that an Euler circuit is also present. However, the inverse is not always true.
03

Final Conclusion

By combining the conditions for both the Euler circuit and Hamiltonian cycle, any graph that fulfills the properties of a Hamiltonian cycle would also contain an Euler circuit, given that a Hamiltonian cycle is a more restrictive condition. Therefore, the graph type would be a simple graph where each node has a degree of at least two and is able to form a Hamilton cycle.

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Most popular questions from this chapter

a) For \(n \in \mathbf{Z}^{+}, n \geq 2\), show that the number of distinct Hamilton cycles in the graph \(K_{n, n}\) is \((1 / 2)(n-1) ! n !\) b) How many different Hamilton paths are there for \(K_{n, n}, n \geq 1\) ?

a) Let \(X=\\{1,2,3,4,5\\}\). Construct the loop-free undirected graph \(G=(V, E)\) as follows: \- \((V)\) : Let each two-element subset of \(X\) represent a vertex in \(G\). \- (E): If \(v_{1}, v_{2} \in V\) correspond to subsets \(\\{a, b\\}\) and \(\\{c, d\\}\), respectively, of \(X\), then draw the edge \(\left\\{v_{1}, v_{2}\right\\}\) in \(G\) if \(\\{a, b\\} \cap\\{c, d\\}=\emptyset\). b) To what graph is \(G\) isomorphic?

Let \(G=(V, E)\) be an undirected graph, where \(|V| \geq 2\). If every induced subgraph of \(G\) is connected, can we identify the graph \(G ?\)

Let \(n=2^{k}\) for \(k \in \mathbf{Z}^{+}\). We use the \(n k\)-bit sequences (of 0 's and 1's) to represent \(1,2,3, \ldots, n\), so that for two consecutive integers \(i, i+1\), the corresponding \(k\)-bit sequences differ in exactly one component. This representation is called a Gray Code. a) For \(k=3\), use a graph model with \(V=\\{000,001,010, \ldots, 111\\}\) to find such a code for \(1,2,3, \ldots, 8\). How is this related to the concept of a Hamilton path? b) Answer part (a) for \(k=4\).

Let \(G=(V, E)\) be a loop-free connected 4-regular planar graph. If \(|E|=16\), how many regions are there in a planar depiction of \(G\) ?
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