On super vertex-graceful unicyclic graphs

Sin Min Lee; Elo Leung; Ho Kuen Ng

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 1, page 1-22
  • ISSN: 0011-4642

Abstract

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A graph G with p vertices and q edges, vertex set V ( G ) and edge set E ( G ) , is said to be super vertex-graceful (in short SVG), if there exists a function pair ( f , f + ) where f is a bijection from V ( G ) onto P , f + is a bijection from E ( G ) onto Q , f + ( ( u , v ) ) = f ( u ) + f ( v ) for any ( u , v ) E ( G ) , Q = { ± 1 , , ± 1 2 q } , if q is even, { 0 , ± 1 , , ± 1 2 ( q - 1 ) } , if q is odd, and P = { ± 1 , , ± 1 2 p } , if p is even, { 0 , ± 1 , , ± 1 2 ( p - 1 ) } , if p is odd. We determine here families of unicyclic graphs that are super vertex-graceful.

How to cite

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Lee, Sin Min, Leung, Elo, and Ng, Ho Kuen. "On super vertex-graceful unicyclic graphs." Czechoslovak Mathematical Journal 59.1 (2009): 1-22. <http://eudml.org/doc/37904>.

@article{Lee2009,
abstract = {A graph $G$ with $p$ vertices and $q$ edges, vertex set $V(G)$ and edge set $E(G)$, is said to be super vertex-graceful (in short SVG), if there exists a function pair $(f, f^+)$ where $f$ is a bijection from $V(G)$ onto $P$, $f^+$ is a bijection from $E(G)$ onto $Q$, $f^+((u, v)) = f(u) + f(v)$ for any $(u, v) \in E(G)$, \[ Q = \{\left\lbrace \begin\{array\}\{ll\} \lbrace \pm 1,\dots , \pm \frac\{1\}\{2\}q\rbrace ,&\text\{if $q$ is even,\}\\ \lbrace 0, \pm 1, \dots , \pm \frac\{1\}\{2\}(q-1)\rbrace ,&\text\{if $q$ is odd,\} \end\{array\}\right.\} \] and \[ P = \{\left\lbrace \begin\{array\}\{ll\} \lbrace \pm 1,\dots , \pm \frac\{1\}\{2\}p\rbrace ,&\text\{if $p$ is even,\}\\ \lbrace 0, \pm 1, \dots , \pm \frac\{1\}\{2\}(p-1)\rbrace ,&\text\{if $p$ is odd.\} \end\{array\}\right.\} \] We determine here families of unicyclic graphs that are super vertex-graceful.},
author = {Lee, Sin Min, Leung, Elo, Ng, Ho Kuen},
journal = {Czechoslovak Mathematical Journal},
keywords = {graceful; edge-graceful; super edge-graceful; super vertex-graceful; amalgamation; trees; unicyclic graphs; graceful; edge-graceful; super edge-graceful; super vertex-graceful; amalgamation; tree; unicyclic graph},
language = {eng},
number = {1},
pages = {1-22},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On super vertex-graceful unicyclic graphs},
url = {http://eudml.org/doc/37904},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Lee, Sin Min
AU - Leung, Elo
AU - Ng, Ho Kuen
TI - On super vertex-graceful unicyclic graphs
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 1
SP - 1
EP - 22
AB - A graph $G$ with $p$ vertices and $q$ edges, vertex set $V(G)$ and edge set $E(G)$, is said to be super vertex-graceful (in short SVG), if there exists a function pair $(f, f^+)$ where $f$ is a bijection from $V(G)$ onto $P$, $f^+$ is a bijection from $E(G)$ onto $Q$, $f^+((u, v)) = f(u) + f(v)$ for any $(u, v) \in E(G)$, \[ Q = {\left\lbrace \begin{array}{ll} \lbrace \pm 1,\dots , \pm \frac{1}{2}q\rbrace ,&\text{if $q$ is even,}\\ \lbrace 0, \pm 1, \dots , \pm \frac{1}{2}(q-1)\rbrace ,&\text{if $q$ is odd,} \end{array}\right.} \] and \[ P = {\left\lbrace \begin{array}{ll} \lbrace \pm 1,\dots , \pm \frac{1}{2}p\rbrace ,&\text{if $p$ is even,}\\ \lbrace 0, \pm 1, \dots , \pm \frac{1}{2}(p-1)\rbrace ,&\text{if $p$ is odd.} \end{array}\right.} \] We determine here families of unicyclic graphs that are super vertex-graceful.
LA - eng
KW - graceful; edge-graceful; super edge-graceful; super vertex-graceful; amalgamation; trees; unicyclic graphs; graceful; edge-graceful; super edge-graceful; super vertex-graceful; amalgamation; tree; unicyclic graph
UR - http://eudml.org/doc/37904
ER -

References

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  1. Cabannis, S., Mitchem, J., Low, R., On edge-graceful regular graphs and trees, Ars Combin. 34 (1992), 129-142. (1992) MR1206556
  2. Gallian, J. A., A dynamic survey of graph labeling, Electronic J. Combin. (2001), 6 1-144. (2001) MR1668059
  3. Keene, J., Simoson, A., Balanced strands for asymmetric, edge-graceful spiders, Ars Combin. 42 (1996), 49-64. (1996) Zbl0856.05087MR1386927
  4. Kuan, Q., Lee, S.-M., Mitchem, J., Wang, A. K., On edge-graceful unicyclic graphs, Congress. Numer. 61 (1988), 65-74. (1988) MR0961638
  5. Lee, L. M., Lee, S.-M., Murty, G., On edge-graceful labelings of complete graphs-solutions of Lo's conjecture, Congress. Numer. 62 (1988), 225-233. (1988) MR0961686
  6. Lee, S.-M., A conjecture on edge-graceful trees, Scientia, Ser. A 3 (1989), 45-57. (1989) Zbl0741.05025MR2309605
  7. Lee, S.-M., New directions in the theory of edge-graceful graphs, Proceedings of the 6th Caribbean Conference on Combinatorics & Computing (1991), 216-231. (1991) 
  8. Lee, S.-M., On strongly indexable graphs and super vertex-graceful graphs, manuscript, . 
  9. Lee, S.-M., Leung, E., On super vertex-graceful trees, Congress. Numer. 167 (2004), 3-26. (2004) Zbl1062.05130MR2122017
  10. Lee, S.-M., Ma, P., Valdes, L., Tong, S.-M., On the edge-graceful grids, Congress. Numer. 154 (2002), 61-77. (2002) Zbl1022.05074MR1980029
  11. Lee, S.-M., Seah, E., Edge-graceful labelings of regular complete k -partite graphs, Congress. Numer. 75 (1990), 41-50. (1990) Zbl0727.05051MR1069161
  12. Lee, S.-M., Seah, E., On edge-gracefulness of the composition of step graphs with null graphs, Combinatorics, Algorithms, and Applications in Society for Industrial and Applied Mathematics (1991), 326-330. (1991) Zbl0741.05060MR1132915
  13. Lee, S.-M., Seah, E., On the edge-graceful ( n , k n ) -multigraphs conjecture, J. Comb. Math. and Comb. Computing 9 (1991), 141-147. (1991) Zbl0735.05072MR1111847
  14. Lee, S.-M., Seah, E., Lo, S. P., On edge-graceful 2-regular graphs, J. Comb. Math. and Comb. Computing 12 (1992), 109-117. (1992) 
  15. Lee, S.-M., Seah, E., Tong, S.-M., On the edge-magic and edge-graceful total graphs conjecture, Congress. Numer. 141 (1999), 37-48. (1999) Zbl0970.05036MR1745223
  16. Lee, S.-M., Seah, E., Wang, P. C., On edge-gracefulness of the kth power graphs, Bull. Inst. Math. Academia Sinica 18 (1990), 1-11. (1990) Zbl0699.05048MR1072825
  17. Lo, S. P., On edge-graceful labelings of graphs, Congress. Numer. 50 (1985), 231-241. (1985) Zbl0597.05054MR0833554
  18. Peng, J., Li, W., Edge-gracefulness of C m × C n , Proceedings of the Sixth Conference of Operations Research Society of China Hong Kong: Global-Link Publishing Company, Changsha (2000), 942-948. (2000) 
  19. Mitchem, J., Simoson, A., On edge-graceful and super-edge-graceful graphs, Ars Combin. 37 (1994), 97-111. (1994) Zbl0805.05074MR1282548
  20. Riskin, A., Wilson, S., Edge graceful labellings of disjoint unions of cycles, Bulletin of the Institute of Combinatorics and its Applications 22 (1998), 53-58. (1998) Zbl0894.05047MR1489867
  21. Schaffer, K., Lee, S.-M., Edge-graceful and edge-magic labelings of Cartesian products of graphs, Congress. Numer. 141 (1999), 119-134. (1999) Zbl0968.05067MR1745230
  22. Shiu, W. C., Lee, S.-M., Schaffer, K., Some k -fold edge-graceful labelings of ( p , p - 1 ) -graphs, J. Comb. Math. and Comb. Computing 38 (2001), 81-95. (2001) Zbl0977.05122MR1853007
  23. Wilson, S., Riskin, A., Edge-graceful labellings of odd cycles and their products, Bulletin of the ICA 24 (1998), 57-64. (1998) Zbl0913.05084MR1638978

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