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arXiv · 2009.13709

Mixed Cages: monotony, connectivity and upper bounds

Gabriela Araujo-Pardo·Claudia de la Cruz·Diego González-Moreno
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Abstract

A \emph{$[z, r; g]$-mixed cage} is a mixed graph $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. %In this paper we study structural properties of mixed cages: Let $n[z,r;g]$ denote the order of a $[z,r;g]$-mixed cage. In this paper we prove that $n[z,r;g]$ is a monotonicity function, with respect of $g$, for $z\in \{1,2\}$, and we use it to prove that the underlying graph of a $[z,r;g]$-mixed cage is 2-connected, for $z\in \{1,2\}$. We also prove that $[z,r;g]$-mixed cages are strong connected. We present bounds of $n[z,r;g]$ and constructions of $[z,r;5]$-mixed graphs and show a $[10,3;5]$-mixed cage of order $50$.

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Gabriela Araujo-Pardo, Claudia de la Cruz, Diego González-Moreno. 2020-09-29. Mixed Cages: monotony, connectivity and upper bounds. https://arxiv.org/abs/2009.13709

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