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A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if for every vertex v ∈ V (G), the subgraph induced by the neighborhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3(More)
The well-known Chvátal–Erd˝ os Theorem states that every graph G of order at least three with α(G) ≤ κ(G) has a hamiltonian cycle, where α(G) and κ(G) are the independence number and the connectivity of G, respectively. Oberly and Sumner [J. Graph Theory 3 (1979), 351–356] have proved that every connected, locally-connected claw-free graph of order at least(More)
The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if C is a hamiltonian cycle of a graph G, then the square of C is called a hamiltonian square of G. In this paper, we characterize all possible forbidden pairs, which implies the containment of a hamiltonian square,(More)
A graph is called 2K2-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every 3 2 -tough 2K2-free graph with at least three vertices has a spanning trail with maximum degree at most 4. In this paper, we confirm this conjecture. We also provide examples for all t < 54 of t-tough graphs that do not(More)
Let G be a simple graph of order n, and let ∆(G) and χ ′ (G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ ′ (G) = ∆(G) or ∆(G) + 1. Following this result, G is called edge-chromatic critical if χ ′ (G) = ∆(G) + 1 and χ ′ (G − e) = ∆(G) for every e ∈ E(G). In 1968, Vizing conjectured that if G is edge-chromatic(More)
Bollobás and Thomason showed that a multigraph of order n and size at least n+ c (c ≥ 1) contains a cycle of length at most 2( n/c + 1) log2 2c . We show in this paper that a multigraph (with no loop) of order n and minimum degree at least 5 contains a chorded cycle (a cycle with a chord) of length at most 300 log2 n. As an application of this result, we(More)
Aiming at the PrefixSpan algorithm produce huge amount of project databases in mining sequence patterns, this paper proposes an Improved PrefixSpan algorithm for Mining Sequential Patterns(IPMSP) algorithm. By avoid produce duplicated project databases with the same prefix pattern through checking the prefix with regard to prefix of the sequence database(More)