Conditions on subgraphs, degrees, and domination for hamiltonian properties of graphs

We present some results concerning hamiltonian properties of recursive circulant graphs in the presence of faulty vertices and/or edges. The recursive circulant graph G(N, d) with d ≥ 2 has vertex set V(G) = {0, 1, …, N - 1} and the edge set E(G) = {(v, w)| ∃ i, 0 ≤ i ≤ ⌈ log d N⌉ - 1, such that v = w + di (mod N)}. When N = cdk where d ≥ 2 and 2 ≤ c ≤ d, G(cdk, d) is regular, node symmetric and can be recursively constructed. G(cdk, d) is a bipartite graph if and only if c is even and d is odd. Let F, the faulty set, be a subset of V(G(cdk, d)) ∪ E(G(cdk, d)). In this paper, we prove that G(cdk, d) - F remains hamiltonian if |F| ≤ deg (G(cdk, d)) - 2 and G(cdk, d) is not bipartite. Moreover, if |F| ≤ deg (G(cdk, d)) - 3 and G(cdk, d) is not a bipartite graph, we prove a more stronger result that for any two vertices u and v in V(G(cdk, d)) - F, there exists a hamiltonian path of G(cdk, d) - F joining u and v.

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The Hamiltonian properties of supergrid graphs

Theoretical Computer Science ◽

10.1016/j.tcs.2015.08.024 ◽

2015 ◽

Vol 602 ◽

pp. 132-148 ◽

Cited By ~ 7

Author(s):

Ruo-Wei Hung ◽

Chih-Chia Yao ◽

Shang-Ju Chan

Keyword(s):

Hamiltonian Properties

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The Hyper-Zagreb Index and Some Properties of Graphs

Handbook of Research on Advanced Applications of Graph Theory in Modern Society - Advances in Computer and Electrical Engineering ◽

10.4018/978-1-5225-9380-5.ch006 ◽

2020 ◽

pp. 120-134 ◽

Cited By ~ 1

Author(s):

Rao Li

Keyword(s):

Topological Index ◽

Sufficient Conditions ◽

Disjoint Paths ◽

Zagreb Index ◽

Connected Graphs ◽

Second Zagreb Index ◽

Vertex Disjoint ◽

Hamiltonian Properties

Let G = (V(G), E(G)) be a graph. The complement of G is denoted by Gc. The forgotten topological index of G, denoted F(G), is defined as the sum of the cubes of the degrees of all the vertices in G. The second Zagreb index of G, denoted M2(G), is defined as the sum of the products of the degrees of pairs of adjacent vertices in G. A graph Gisk-Hamiltonian if for all X ⊂V(G) with|X| ≤ k, the subgraph induced byV(G) - Xis Hamiltonian. Clearly, G is 0-Hamiltonian if and only if G is Hamiltonian. A graph Gisk-path-coverableifV(G) can be covered bykor fewer vertex-disjoint paths. Using F(Gc) and M2(Gc), Li obtained several sufficient conditions for Hamiltonian and traceable graphs (Rao Li, Topological Indexes and Some Hamiltonian Properties of Graphs). In this chapter, the author presents sufficient conditions based upon F(Gc) and M2(Gc)for k-Hamiltonian, k-edge-Hamiltonian, k-path-coverable, k-connected, and k-edge-connected graphs.

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