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ABSTRACT: The square
$G^2$
of a graph
$G$
is defined on the vertex set
$V(G)$
of
$G$
such that any two vertices with distance at most two in
$G$
are linked by an edge. In this paper, the chromatic number and equitable chromatic number of the square
$S^2(n,k)$
of Sierpiński graph
$S(n,k)$
are studied. It is obtained that
$\chi (S^2(n,k))=\chi _{=}(S^2(n,k))=k+1$
for
$n\ge 2$
and
$k\ge 2$
. Graphs and Combinatorics 06/2014; · 0.35 Impact Factor

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ABSTRACT: A proper tcoloring of a graph G is a mapping
${\varphi: V(G) \rightarrow [1, t]}$
such that
${\varphi(u) \neq \varphi(v)}$
if u and v are adjacent vertices, where t is a positive integer. The chromatic number of a graph G, denoted by
${\chi(G)}$
, is the minimum number of colors required in any proper coloring of G. A linear tcoloring of a graph is a proper tcoloring such that the graph induced by the vertices of any two color classes is the union of vertexdisjoint paths. The linear chromatic number of a graph G, denoted by
${lc(G)}$
, is the minimum t such that G has a linear tcoloring. In this paper, the linear tcolorings of Sierpińskilike graphs S(n, k),
${S^+(n, k)}$
and
${S^{++}(n, k)}$
are studied. It is obtained that
${lc(S(n, k))= \chi (S(n, k)) = k}$
for any positive integers n and k,
${lc(S^+(n, k)) = \chi(S^+(n, k)) = k}$
and
${lc(S^{++}(n, k)) = \chi(S^{++}(n, k)) = k}$
for any positive integers
${n \geq 2}$
and
${k \geq 3}$
. Furthermore, we have determined the number of paths and the length of each path in the subgraph induced by the union of any two color classes completely. Graphs and Combinatorics 01/2014; · 0.35 Impact Factor

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ABSTRACT: In [23], Klavzar and Milutinovic (1997) proved that there exist at most two different shortest paths between any two vertices in Sierpinski graphs S"k^n, and showed that the number of shortest paths between any fixed pair of vertices of S"k^n can be computed in O(n). An almostextreme vertex of S"k^n, which was introduced in Klavzar and Zemljic (2013) [27], is a vertex that is either adjacent to an extreme vertex or incident to an edge between two subgraphs of S"k^n isomorphic to S"k^n^^1. In this paper, we completely determine the set S"u={v@?V(S"k^n):there exist two shortest u,vpaths in S"k^n}, where u is any almostextreme vertex of S"k^n. Discrete Applied Mathematics 01/2014; 162:314321. · 0.72 Impact Factor

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ABSTRACT: A mapping @f from V(G) to {1,2,...,t} is called a pathtcoloring of a graph G if each G[@f^^1(i)], for 1@?i@?t, is a linear forest. The vertex linear arboricity of a graph G, denoted by vla(G), is the minimum t for which G has a path tcoloring. Graphs S[n,k] are obtained from the Sierpinski graphs S(n,k) by contracting all edges that lie in no induced K"k. In this paper, the hamiltonicity and path tcoloring of Sierpinskilike graphs S(n,k), S^+(n,k), S^+^+(n,k) and graphs S[n,k] are studied. In particular, it is obtained that vla(S(n,k))=vla(S[n,k])=@?k/2@? for k>=2. Moreover, the numbers of edge disjoint Hamiltonian paths and Hamiltonian cycles in S(n,k), S^+(n,k) and S^+^+(n,k) are completely determined, respectively. Discrete Applied Mathematics 08/2012; 160(12):1822–1836. · 0.72 Impact Factor

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ABSTRACT: A linear kforest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear karboricity of G, denoted by lak(G), is the minimum number of linear kforests needed to partition the edge set E(G) of G. In the case where the lengths of paths are not restricted, we then have the linear arboricity of G, denoted by la(G). In this paper, we obtain the exact value of the linear (n−1)arboricity of any balanced complete npartite graph Kn(m). Discrete Applied Mathematics. 07/2010;

Discrete Applied Mathematics. 01/2010; 158:15461550.

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ABSTRACT: A linear kforest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear karboricity of G, denoted by lak(G), is the minimum number of linear kforests needed to partition the edge set E(G) of G. In the case where the lengths of paths are not restricted, we then have the linear arboricity of G, denoted by la(G). In this paper, we obtain the exact value of the linear (n−1)arboricity of any balanced complete npartite graph Kn(m). Discrete Applied Mathematics  DAM. 01/2010; 158(14):15461550.

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ABSTRACT: The minimum and maximum values of Randić index for the chemical trees with given order and number of pendent vertex were obtained by P. Hansen and H. Mélot [J. Chem. Inf. Comput. Sci. 43, 1–14 (2003)], and the corresponding chemical trees are characterized. But there is a serious mistake in their proof. A correct proof of the theorem is given in this paper. A new chemical tree T ' is obtained by either adding pendent edges to the chemical tree T or subdividing it, and the minimum value of R(T) is calculated using the minimum value of R(T ' )R(T). Journal of Tianjin Normal University. Natural Science Edition. 01/2009; 29(4).