# On k-Path Pancyclic Graphs

• Volume: 35, Issue: 2, page 271-281
• ISSN: 2083-5892

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## Abstract

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For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G is k-path pancyclic

## How to cite

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Zhenming Bi, and Ping Zhang. "On k-Path Pancyclic Graphs." Discussiones Mathematicae Graph Theory 35.2 (2015): 271-281. <http://eudml.org/doc/271089>.

@article{ZhenmingBi2015,
abstract = {For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G is k-path pancyclic},
author = {Zhenming Bi, Ping Zhang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Hamiltonian; panconnected; pancyclic; path Hamiltonian; path pancyclic; Hamiltonian graph; panconnected graph; pancyclic graph; path-Hamiltonian graph; path-pancyclic graph},
language = {eng},
number = {2},
pages = {271-281},
title = {On k-Path Pancyclic Graphs},
url = {http://eudml.org/doc/271089},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Zhenming Bi
AU - Ping Zhang
TI - On k-Path Pancyclic Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 2
SP - 271
EP - 281
AB - For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G is k-path pancyclic
LA - eng
KW - Hamiltonian; panconnected; pancyclic; path Hamiltonian; path pancyclic; Hamiltonian graph; panconnected graph; pancyclic graph; path-Hamiltonian graph; path-pancyclic graph
UR - http://eudml.org/doc/271089
ER -

## References

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10. [10] C.St.J.A. Nash-Williams, Problem No. 48, in: Theory of Graphs and its Applica- tions, (Academic Press, New York 1968), 367.
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12. [12] O. Ore, Hamilton connected graphs, J. Math. Pures Appl. 42 (1963) 21-27. Zbl0106.37103
13. [13] B. Randerath, I. Schiermeyer, M. Tewes and L. Volkmann, Vertex pancyclic graphs, Discrete Appl. Math. 120 (2002) 219-237. doi:10.1016/S0166-218X(01)00292-X[Crossref] Zbl1001.05070
14. [14] M. Sekanina, On an ordering of the set of vertices of a connected graph, Publ. Fac. Sci. Univ. Brno 412 (1960) 137-142.
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