On (k1A1, k2A2, k3A3)-Edge Colourings in Graphs and Generalized Jacobsthal Numbers

Piejko, Krzysztof; Trojnar-Spelina, Lucyna

doi:10.2478/amsil-2025-0006

Abstract

In this paper we introduce a new kind of generalized Jacobsthal numbers in a distance sense. We give the identities and matrix representations for them and their connections with the Fibonacci and the Pell numbers. We also describe the interpretations of these numbers in terms of some kind of (k₁A₁, k₂A₂, k₃A₃)-edge colouring and quasi colouring.

References

U. Bednarz, D. Bród, K. Piejko, and A. Włoch, (k, α_n−1)-Fibonacci numbers and P_k-matchings in multigraphs, Ars Combin. 111 (2013), 354–355.
Search in Google Scholar Back to article
U. Bednarz, D. Bród, and M. Wołowiec-Musiał, On two types of (2, k)-distance Lucas numbers, Ars Combin. 115 (2014), 467–479.
Search in Google Scholar Back to article
U. Bednarz, A. Włoch, and M. Wołowiec-Musiał, Distance Fibonacci numbers, their interpretations and matrix generators, Comment. Math. 53 (2013), no. 1, 35–46.
Search in Google Scholar Back to article
C. Berge, Principles of Combinatorics, Academic Press, New York-London, 1971.
Search in Google Scholar Back to article
D. Bród, K. Piejko, and I. Włoch, Distance Fibonacci numbers, distance Lucas numbers and their applications, Ars Combin. 112 (2013), 397–409.
Search in Google Scholar Back to article
R. Diestel, Graph Theory, Springer-Verlag, Heidelberg-New York, 2005.
Search in Google Scholar Back to article
J. Ercolano, Matrix generator of Pell sequence, Fibonacci Quart. 17 (1979), no. 1, 71–77.
Search in Google Scholar Back to article
E. Kiliç, The generalized order-k Fibonacci-Pell sequence by matrix methods, J. Comput. Appl. Math. 209 (2007), no. 2, 133–145.
Search in Google Scholar Back to article
E. Kiliç, On the usual Fibonacci and generalized order-k Pell numbers, Ars Combin. 88 (2008), 33–45.
Search in Google Scholar Back to article
E. Kiliç and D. Taşci, The generalized Binet formula, representation and sums of the generalized order-k Pell numbers, Taiwanese J. Math. 10 (2006), no. 6, 1661–1670.
Search in Google Scholar Back to article
F. Koken and D. Bozkurt, On the Jacobsthal numbers by matrix methods, Int. J. Contemp. Math. Sci. 3 (2008), no. 13, 605–614.
Search in Google Scholar Back to article
K. Piejko, Extremal trees with respect to numbers of (A, B, 2C)-edge colourings, J. Appl. Math. 2015, Art. ID 463650, 5 pp.
Search in Google Scholar Back to article
K. Piejko, On k-distance Pell numbers and ((k − 1)A, (k − 1)B, kC)-edge coloured graphs, Ars Combin. 139 (2018), 197–215.
Search in Google Scholar Back to article
K. Piejko, On the number of (A, B, 2C)-edge colourings in graphs, Ars Combin. 139 (2018), 27–41.
Search in Google Scholar Back to article
K. Piejko and I. Włoch, On k-distance Pell numbers in 3-edge-coloured graphs, J. Appl. Math. 2014, Art. ID 428020, 6 pp.
Search in Google Scholar Back to article
A. Szynal-Liana and I. Włoch, On distance Pell numbers and their connections with Fibonacci numbers, Ars Combin. 113A (2014), 65–75.
Search in Google Scholar Back to article
L. Trojnar-Spelina and I. Włoch, On Pell numbers and (k₁A₁, k₂A₂, k₃A₃)-edge colouring in graphs, Ars Combin. 125 (2016), 183–191.
Search in Google Scholar Back to article
I. Włoch, On generalized Pell numbers and their graph representations, Comment. Math. 48 (2008), no. 2, 169–175.
Search in Google Scholar Back to article
I. Włoch, U. Bednarz, D. Bród, A. Włoch, and M. Wołowiec-Musiał, On a new type of distance Fibonacci numbers, Discrete Appl. Math. 161 (2013), no. 16–17, 2695–2701.
Search in Google Scholar Back to article
A. Włoch and M. Wołowiec-Musiał, Generalized Pell numbers and some relations with Fibonacci numbers, Ars Combin. 109 (2013), 391–403.
Search in Google Scholar Back to article

On (k1A1, k2A2, k3A3)-Edge Colourings in Graphs and Generalized Jacobsthal Numbers

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