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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.56 no.1 Bogotá Jan./June 2022 Epub Feb 04, 2023
https://doi.org/10.15446/recolma.v56n1.105617
Original article
Sizes of flats of cycle matroids of complete graphs
Los tamaños de los cerrados de la matroide gráfica del grafo completo
1 University of the Witwatersrand, Johannesburg, South Africa
We show that the problem of counting the number of flats of size k for a cycle matroid of a complete graph is equivalent to the problem of counting the number of partitions of an integer k into triangular numbers. In addition, we give some values of k such that there is no flat of size k in a cycle matroid of a complete graph of order n. Finally, we give a minimum bound for the number of values, k, for which there are no flats of size k in the given cycle matroid.
Keywords: Compositions; cycle matroid; flats; triangular number partitions; bad colouring
Demostraremos que el problema de contar los conjuntos cerrados de tamaño k de la matroide gráfica de un grafo completo es equivalente al problema de contar las particiones de un entero k en números triangulares. Adicionalmente, daremos unos valores de k tales que no existe ningún cerrado de tamaño k en la matroide gráfica de un grafo completo de orden n. Finalmente, daremos una cota inferior para el número de valores k para los cuales no existe ningún cerrado de tamaño k en la matroide gráfica.
Palabras clave: Composiciones; matroide; particiones de números triangulares
References
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6. T. Trotter, Some identities for the Triangular Numbers, Journal of Recreational Mathematics 6 (1973), no. 2, 127-135. [ Links ]
Received: September 16, 2020; Accepted: May 27, 2022
This is an open-access article distributed under the terms of the Creative Commons Attribution License
