Special Graceful Labelings of Irregular Fences and Lobsters

Christian Barrientos

doi:10.32323/ujma.520682

EN

Special Graceful Labelings of Irregular Fences and Lobsters

Abstract

Irregular fences are subgraphs of $P_m \times P_n$ formed with $m$ copies of $P_n$ in such a way that two consecutive copies of $P_n$ are connected with one or two edges; if two edges are used, then they are located in levels separated an odd number of units. We prove here that any of these fences admits a special kind of graceful labeling, called $\alpha$-labeling. We show that there is a huge variety of this type of fences presenting a closed formula to determine the number of them that can be built on the grid $[1,m] \times [1, n]$. If only one edge is used to connect any pair of consecutive copies of $P_n$, the resulting graph is a tree. We use the $\alpha$-labelings of this type of fences to construct and label a subfamily of lobsters, partially answering the long standing conjecture of Bermond that states that all lobsters are graceful. The final labeling of the lobsters presented here is not only graceful, it is an $\alpha$-labeling, therefore they can be used to produce new graceful trees.

Keywords

Fence,Graceful labelling,Lobster

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Christian Barrientos ^*
0000-0003-2838-8687
United States

Publication Date

March 20, 2019

Submission Date

February 1, 2019

Acceptance Date

March 6, 2019

Published in Issue

Year 2019 Volume: 2 Number: 1

DOI

https://doi.org/10.32323/ujma.520682

IZ

https://izlik.org/JA49KE28FG

APA

Barrientos, C. (2019). Special Graceful Labelings of Irregular Fences and Lobsters. Universal Journal of Mathematics and Applications, 2(1), 1-10. https://doi.org/10.32323/ujma.520682

AMA

1.Barrientos C. Special Graceful Labelings of Irregular Fences and Lobsters. Univ. J. Math. Appl. 2019;2(1):1-10. doi:10.32323/ujma.520682

Chicago

Barrientos, Christian. 2019. “Special Graceful Labelings of Irregular Fences and Lobsters”. Universal Journal of Mathematics and Applications 2 (1): 1-10. https://doi.org/10.32323/ujma.520682.

EndNote

Barrientos C (March 1, 2019) Special Graceful Labelings of Irregular Fences and Lobsters. Universal Journal of Mathematics and Applications 2 1 1–10.

IEEE

[1]C. Barrientos, “Special Graceful Labelings of Irregular Fences and Lobsters”, Univ. J. Math. Appl., vol. 2, no. 1, pp. 1–10, Mar. 2019, doi: 10.32323/ujma.520682.

ISNAD

Barrientos, Christian. “Special Graceful Labelings of Irregular Fences and Lobsters”. Universal Journal of Mathematics and Applications 2/1 (March 1, 2019): 1-10. https://doi.org/10.32323/ujma.520682.

JAMA

1.Barrientos C. Special Graceful Labelings of Irregular Fences and Lobsters. Univ. J. Math. Appl. 2019;2:1–10.

MLA

Barrientos, Christian. “Special Graceful Labelings of Irregular Fences and Lobsters”. Universal Journal of Mathematics and Applications, vol. 2, no. 1, Mar. 2019, pp. 1-10, doi:10.32323/ujma.520682.

Vancouver

1.Christian Barrientos. Special Graceful Labelings of Irregular Fences and Lobsters. Univ. J. Math. Appl. 2019 Mar. 1;2(1):1-10. doi:10.32323/ujma.520682