New Advances in Kotzig's Conjecture

Christian BARRİENTOS; Sarah MİNİON

doi:https://doi.org/10.33401/fujma.563563

Makale detayları

New Advances in Kotzig's Conjecture

Yazarlar:

Christian BARRİENTOS | Sarah MİNİON

Yayıncı:

Fuat USTA

Ana Başlıklar:

Matematik

Dergi:

Fundamental Journal of Mathematics and Applications 2019 Vol.2 Issue 2, pp.186-194

Belge Tipi:

Makale

Yayın Tarihi:

12/2019

Belge Dili:

İngilizce

Document ID (Org No.):

2000422175

pISSN:

2645-8845

DOI:

https://doi.org/10.33401/fujma.563563

Özet:

 


In 1973 Kotzig conjectures that the complete graph K2n+1K2n+1K_{2n+1} can be cyclically decomposed into 2n+12n+12n+1 copies of any tree of size nnn. Rosa proved that this decomposition exists if and only if there exists a ρρ\rho-labeling of the tree. In this work we prove that if T′T′T' is a graceful tree, then any tree TTT obtained from T′T′T' by attaching a total of k≥1k≥1k \geq 1 pendant vertices to any collection of rrr vertices of T′T′T', where 1≤r≤k1≤r≤k1 \leq r \leq k, admits a ρρ\rho-labeling. As a consequence of this result, many new families of trees with this kind of labeling are produced, which indicates the strong potential of this result. Moreover, the technique used to prove this result, gives us an indication of how to determine whether a given tree of size nnn decomposes the complete graph K2n+1K2n+1K_{2n+1}. We also prove the existence of a ρρ\rho-labeling for two subfamilies of lobsters and present a method to produce ρρ\rho-labeled trees attaching pendant vertices and pendant copies of the path P3P3P_3 to some of the vertices of any graceful tree.\\
In addition, for any given tree TTT, we use bipartite labelings to show that this tree is a spanning tree of a graph GGG that admits an αα\alpha-labeling. This is not a new result; however, the construction presented here optimizes (reduces) the size of GGG with respect to all the similar results that we found in the literature.

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Makale detayları

New Advances in Kotzig's Conjecture

Yazarlar: Christian BARRİENTOS | Sarah MİNİON

Yayıncı: Fuat USTA

Ana Başlıklar: Matematik

Dergi: Fundamental Journal of Mathematics and Applications 2019 Vol.2 Issue 2, pp.186-194

Belge Tipi: Makale

Yayın Tarihi: 12/2019

Belge Dili: İngilizce

Document ID (Org No.): 2000422175

pISSN: 2645-8845

DOI: https://doi.org/10.33401/fujma.563563

Sayı

Yazarlar:

Christian BARRİENTOS | Sarah MİNİON

Yayıncı:

Fuat USTA

Ana Başlıklar:

Matematik

Dergi:

Fundamental Journal of Mathematics and Applications 2019 Vol.2 Issue 2, pp.186-194

Belge Tipi:

Makale

Yayın Tarihi:

12/2019

Belge Dili:

İngilizce

Document ID (Org No.):

2000422175

pISSN:

2645-8845

DOI:

https://doi.org/10.33401/fujma.563563