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Spanning caterpillars with bounded diameter

100%
Faudree R., Gould R., Jacobson M., Lesniak L.
Discussiones Mathematicae Graph Theory
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1995
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tom 15
|
nr 2
111-118
EN
A caterpillar is a tree with the property that the vertices of degree at least 2 induce a path. We show that for every graph G of order n, either G or G̅ has a spanning caterpillar of diameter at most 2 log n. Furthermore, we show that if G is a graph of diameter 2 (diameter 3), then G contains a spanning caterpillar of diameter at most $cn^{3/4}$ (at most n).
2
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Recognizable colorings of graphs

100%
Chartrand G., Lesniak L., VanderJagt D., Zhang P.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 1
35-57
EN
Let G be a connected graph and let c:V(G) → {1,2,...,k} be a coloring of the vertices of G for some positive integer k (where adjacent vertices may be colored the same). The color code of a vertex v of G (with respect to c) is the ordered (k+1)-tuple code(v) = (a₀,a₁,...,aₖ) where a₀ is the color assigned to v and for 1 ≤ i ≤ k, $a_i$ is the number of vertices adjacent to v that are colored i. The coloring c is called recognizable if distinct vertices have distinct color codes and the recognition number rn(G) of G is the minimum positive integer k for which G has a recognizable k-coloring. Recognition numbers of complete multipartite graphs are determined and characterizations of connected graphs of order n having recognition numbers n or n-1 are established. It is shown that for each pair k,n of integers with 2 ≤ k ≤ n, there exists a connected graph of order n having recognition number k. Recognition numbers of cycles, paths, and trees are investigated.
3
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Linear forests and ordered cycles

76%
Chen G., Faudree R., Gould R., Jacobson M., Lesniak L., Pfender F.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 3
359-372
EN
A collection $L = P¹ ∪ P² ∪ ... ∪ P^t$ (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V(L)| = k is called a (k,t,s)-linear forest. A graph G is (k,t,s)-ordered if for every (k,t,s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k,t,s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k,t,s)-ordered hamiltonian.
4
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On longest paths in connected graphs

62%
Lesniak L.
Fundamenta Mathematicae
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1974-1975
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tom 86
|
nr 3
283-286
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