2,810 research outputs found

    Path (or cycle)-trees with Graph Equations involving Line and Split Graphs

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    H-trees generalizes the existing notions of trees, higher dimensional trees and k-ctrees. The characterizations and properties of both Pk-trees for k at least 4 and Cn-trees for n at least 5 and their hamiltonian property, dominations, planarity, chromatic and b-chromatic numbers are established. The conditions under which Pk-trees for k at least 3 (resp. Cn-trees for n at least 4), are the line graphs are determined. The relationship between path-trees and split graphs are developed

    On hamiltonian colorings of block graphs

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    A hamiltonian coloring c of a graph G of order p is an assignment of colors to the vertices of G such that D(u,v)+∣c(u)−c(v)∣≥p−1D(u,v)+|c(u)-c(v)|\geq p-1 for every two distinct vertices u and v of G, where D(u,v) denoted the detour distance between u and v. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we present a lower bound for the hamiltonian chromatic number of block graphs and give a sufficient condition to achieve the lower bound. We characterize symmetric block graphs achieving this lower bound. We present two algorithms for optimal hamiltonian coloring of symmetric block graphs.Comment: 12 pages, 1 figure. A conference version appeared in the proceedings of WALCOM 201

    Parity balance of the ii-th dimension edges in Hamiltonian cycles of the hypercube

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    Let n≥2n\geq 2 be an integer, and let i∈{0,...,n−1}i\in\{0,...,n-1\}. An ii-th dimension edge in the nn-dimensional hypercube QnQ_n is an edge v1v2{v_1}{v_2} such that v1,v2v_1,v_2 differ just at their ii-th entries. The parity of an ii-th dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its vertex ignoring the ii-th entry. We prove that the number of ii-th dimension edges appearing in a given Hamiltonian cycle of QnQ_n with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in QnQ_n contains two opposite edges in a 4-cycle. We prove this conjecture for n≤7n \le 7, and for any Hamiltonian cycle containing more than 2n−22^{n-2} edges in the same dimension. This bound is finally improved considering the equi-independence number of Qn−1Q_{n-1}, which is a concept introduced in this paper for bipartite graphs

    Chromatic roots and minor-closed families of graphs

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    Given a minor-closed class of graphs G\mathcal{G}, what is the infimum of the non-trivial roots of the chromatic polynomial of G∈GG \in \mathcal{G}? When G\mathcal{G} is the class of all graphs, the answer is known to be 32/2732/27. We answer this question exactly for three minor-closed classes of graphs. Furthermore, we conjecture precisely when the value is larger than 32/2732/27.Comment: 18 pages, 5 figure
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