1. SET-SEQUENTIAL LABELINGS OF ODD TREES.
- Author
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ECKELS, EMILY N., GYŐRI, ERVIN, JUNSHENG LIU, and NASIR, SOHAIB
- Subjects
- *
BIPARTITE graphs , *GRAPH labelings , *TREES , *GRAPH coloring , *CATERPILLARS - Abstract
A tree T on 2n 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 2n 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 Fn2 in a particular way; in the process, we provide a substantial clarification of a proof of a theorem that partitions Fn2 from a paper [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]. Finally, we put forward a result on bipartite graphs that is a modification of a theorem in the aforementioned paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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