# Heavy subgraph pairs for traceability of block-chains

Binlong Li; Hajo Broersma; Shenggui Zhang

Discussiones Mathematicae Graph Theory (2014)

- Volume: 34, Issue: 2, page 287-307
- ISSN: 2083-5892

## Access Full Article

top## Abstract

top## How to cite

topBinlong Li, Hajo Broersma, and Shenggui Zhang. "Heavy subgraph pairs for traceability of block-chains." Discussiones Mathematicae Graph Theory 34.2 (2014): 287-307. <http://eudml.org/doc/268015>.

@article{BinlongLi2014,

abstract = {A graph is called traceable if it contains a Hamilton path, i.e., a path containing all its vertices. Let G be a graph on n vertices. We say that an induced subgraph of G is o−1-heavy if it contains two nonadjacent vertices which satisfy an Ore-type degree condition for traceability, i.e., with degree sum at least n−1 in G. A block-chain is a graph whose block graph is a path, i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least one cut vertex and exactly two end-blocks. Obviously, every traceable graph is a block-chain, but the reverse does not hold. In this paper we characterize all the pairs of connected o−1-heavy graphs that guarantee traceability of block-chains. Our main result is a common extension of earlier work on degree sum conditions, forbidden subgraph conditions and heavy subgraph conditions for traceability},

author = {Binlong Li, Hajo Broersma, Shenggui Zhang},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {o−1-heavy subgraph; block-chain traceable graph; Ore-type condition; forbidden subgrap; -heavy subgraph; forbidden subgraph},

language = {eng},

number = {2},

pages = {287-307},

title = {Heavy subgraph pairs for traceability of block-chains},

url = {http://eudml.org/doc/268015},

volume = {34},

year = {2014},

}

TY - JOUR

AU - Binlong Li

AU - Hajo Broersma

AU - Shenggui Zhang

TI - Heavy subgraph pairs for traceability of block-chains

JO - Discussiones Mathematicae Graph Theory

PY - 2014

VL - 34

IS - 2

SP - 287

EP - 307

AB - A graph is called traceable if it contains a Hamilton path, i.e., a path containing all its vertices. Let G be a graph on n vertices. We say that an induced subgraph of G is o−1-heavy if it contains two nonadjacent vertices which satisfy an Ore-type degree condition for traceability, i.e., with degree sum at least n−1 in G. A block-chain is a graph whose block graph is a path, i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least one cut vertex and exactly two end-blocks. Obviously, every traceable graph is a block-chain, but the reverse does not hold. In this paper we characterize all the pairs of connected o−1-heavy graphs that guarantee traceability of block-chains. Our main result is a common extension of earlier work on degree sum conditions, forbidden subgraph conditions and heavy subgraph conditions for traceability

LA - eng

KW - o−1-heavy subgraph; block-chain traceable graph; Ore-type condition; forbidden subgrap; -heavy subgraph; forbidden subgraph

UR - http://eudml.org/doc/268015

ER -

## References

top- [1] P. Bedrossian, G. Chen, and R.H. Schelp, A generalization of Fan’s condition for hamiltonicity, pancyclicity, and hamiltonian connectedness, Discrete Math. 115 (1993) 39-50. doi:10.1016/0012-365X(93)90476-A[Crossref] Zbl0773.05075
- [2] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer Graduate Texts in Mathe- matics 244 (2008).
- [3] H.J. Broersma, Z. Ryj´aˇcek and I. Schiermeyer, Dirac’s minimum degree condition restricted to claws, Discrete Math. 167/168 (1997) 155-166. doi:10.1016/S0012-365X(96)00224-5[Crossref]
- [4] H.J. Broersma and H.J. Veldman, Restrictions on induced subgraphs ensuring hamil- tonicity or pancyclicity of K1,3-free graphs, in: R. Bodendiek (Ed.) Contemporary Methods in Graph Theory, (BI Wissenschaftsverlag, Mannheim-Wien-Zfirich, 1990) 181-194.
- [5] R. Čada, Degree conditions on induced claws, Discrete Math. 308 (2008) 5622-5631. doi:10.1016/j.disc.2007.10.026[Crossref][WoS] Zbl1175.05069
- [6] D. Duffus, R.J. Gould and M.S. Jacobson, Forbidden subgraphs and the hamiltonian theme, in: The Theory and Applications of Graphs (Kalamazoo, Mich. 1980, Wiley, New York, 1981) 297-316.
- [7] R.J. Faudree and R.J. Gould, Characterizing forbidden pairs for Hamiltonian prop- erties, Discrete Math. 173 (1997) 45-60. doi:10.1016/S0012-365X(96)00147-1[Crossref]
- [8] H. Fleischner, The square of every two-connected graph is hamiltonian, J. Combin. Theory (B) 16 (1974) 29-34. doi:10.1016/0095-8956(74)90091-4[Crossref] Zbl0256.05121
- [9] B. Li, H.J. Broersma and S. Zhang, Forbidden subgraph pairs for traceability of block-chains, Electron. J. Graph Theory Appl. 1 (2013) 1-10. Zbl1306.05205
- [10] B. Li, Z. Ryjáček, Y.Wang and S. Zhang, Pairs of heavy subgraphs for Hamiltonicity of 2-connected graphs, SIAM J. Discrete Math. 26 (2012) 1088-1103. doi:10.1137/11084786X[WoS] Zbl1256.05052
- [11] B. Li and S. Zhang, On traceability of claw-o−1-heavy graphs (2013). arXiv:1303.0991v1

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.