Geodesic
Set-sequential labelings of odd trees
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 151-170 Cet article a éte moissonné depuis la source Library of Science

Voir la notice de l'article

A tree T on 2^n vertices is called set-sequential if the elements in V(T)∪ E(T) can be labeled with distinct nonzero (n+1)-dimensional 01-vectors such that the vector labeling each edge is the component-wise sum modulo 2 of the labels of the endpoints. It has been conjectured that all trees on 2^n vertices with only odd degree are set-sequential (the “Odd Tree Conjecture”), and in this paper, we present progress toward that conjecture. We show that certain kinds of caterpillars (with restrictions on the degrees of the vertices, but no restrictions on the diameter) are set-sequential. Additionally, we introduce some constructions of new set-sequential graphs from smaller set-sequential bipartite graphs (not necessarily odd trees). We also make a conjecture about pairings of the elements of 𝔽_2^n in a particular way; in the process, we provide a substantial clarification of a proof of a theorem that partitions 𝔽_2^n from a paper [Coloring vertices and edges of a graph by nonempty subsets of a set, European J. Combin. 32 (2011) 533–537] by Balister et al. Finally, we put forward a result on bipartite graphs that is a modification of a theorem in the aforementioned paper.
Keywords: trees, coloring graphs by sets, caterpillars 
@article{DMGT_2024_44_1_a7,
     author = {Eckels, Emily and Gy\H{o}ri, Ervin and Liu, Junsheng and Nasir, Sohaib},
     title = {Set-sequential labelings of odd trees},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {151--170},
     year = {2024},
     volume = {44},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2024_44_1_a7/}
}
TY  - JOUR
AU  - Eckels, Emily
AU  - Győri, Ervin
AU  - Liu, Junsheng
AU  - Nasir, Sohaib
TI  - Set-sequential labelings of odd trees
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2024
SP  - 151
EP  - 170
VL  - 44
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DMGT_2024_44_1_a7/
LA  - en
ID  - DMGT_2024_44_1_a7
ER  - 
%0 Journal Article
%A Eckels, Emily
%A Győri, Ervin
%A Liu, Junsheng
%A Nasir, Sohaib
%T Set-sequential labelings of odd trees
%J Discussiones Mathematicae. Graph Theory
%D 2024
%P 151-170
%V 44
%N 1
%U http://geodesic.mathdoc.fr/item/DMGT_2024_44_1_a7/
%G en
%F DMGT_2024_44_1_a7
Eckels, Emily; Győri, Ervin; Liu, Junsheng; Nasir, Sohaib. Set-sequential labelings of odd trees. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 151-170. http://geodesic.mathdoc.fr/item/DMGT_2024_44_1_a7/

[1] B.D. Acharya and S.M. Hegde, Set sequential graphs, Nat. Acad. Sci. Lett. 8 (1985) 387–390.

[2] K. Abhishek, Set-valued graphs II}, J. Fuzzy Set Valued Anal. 2013 (2013) 1–16.

[3] K. Abhishek and G.K. Agustine, Set-valued graphs, J. Fuzzy Set Valued Anal. 2012 (2012) 1–17.

[4] P.N. Balister, E. Győri and R.H. Schelp, Coloring vertices and edges of a graph by nonempty subsets of a set, European J. Combin. 32 (2011) 533–537. https://doi.org/10.1016/j.ejc.2010.11.008

[5] L. Golowich and C. Kim, New classes of set-sequential trees, Discrete Math. 343 (2020) 111741. https://doi.org/10.1016/j.disc.2019.111741

[6] S.M. Hegde, Set colorings of graphs, European J. Combin. 30 (2009) 986–995. https://doi.org/10.1016/j.ejc.2008.06.005

[7] A.R. Mehta, G.R. Vijayakumar and S. Arumugam, A note on ternary sequences of strings of 0 and 1, AKCE Int. J. Graphs Comb. 5 (2008) 175–179. https://doi.org/10.1080/09728600.2008.12088862