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All Journal Indonesian Journal of Combinatorics
Barrientos, Christian
University of South Florida
Computer Science & IT Decision Sciences, Operations Research & Management

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Zonal Labeling of Graphs Barrientos, Christian; Minion, Sarah
Indonesian Journal of Combinatorics Vol 8, No 2 (2024)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.19184/ijc.2024.8.2.4

Abstract

A planar graph is said to be zonal when is possible to label its vertices with the nonzero elements of ℤ3, in such a way that the sum of the labels of the vertices on the boundary of each zone is 0 in ℤ3. In this work we present some conditions that guarantee the existence of a zonal labeling for a number of families of graphs such as unicyclic and outerplanar, including the family of bipartite graphs with connectivity at least 2 whose stable sets have the same cardinality; additionally, we prove that when any edge of a zonal graph is subdivided twice, the resulting graph is zonal as well. Furthermore, we prove that the Cartesian product G × P2 is zonal, when G is a tree, a unicyclic graph, or certain variety of outerplanar graphs. Besides these results, we determine the number of different zonal labelings of the cycle Cn.
The number of spanning trees of cyclic snakes Barrientos, Christian
Indonesian Journal of Combinatorics Vol 9, No 1 (2025)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.19184/ijc.2024.9.1.3

Abstract

A cyclic snake is a connected graph formed by connecting, by means of vertex amalgamation, a certain number of copies of the cycle Cn, in such a way that the i-th copy of Cn is connected with the (i+1)-th copy, resulting in a graph with maximum degree 4. Spanning trees of this type of graph can be easily found, but finding the number of nonisomorphic spanning trees of a given cyclic snake is a more challenging problem. In this work, we investigate the number of cyclic snakes formed with k copies of Cn, the number of spanning trees of any given cyclic snake. We also classified these trees according to their diameters. Finally, we study the morphology of the trees associated to the snakes where the distance between cut-vertices is a constant.
Co-Authors Minion, Sarah