# Girth, minimum degree, independence, and broadcast independence

Document Type : Original paper

Authors

1 LIRMM

2 89069 Ulm, Germay

Abstract

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to \mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number $\alpha_b(G)$ of $G$ is the largest weight $\sum_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. It is known that $\alpha(G)\leq \alpha_b(G)\leq 4\alpha(G)$ for every connected graph $G$, where $\alpha(G)$ is the independence number of $G$. If $G$ has girth $g$ and minimum degree $\delta$, we show that $\alpha_b(G)\leq 2\alpha(G)$ provided that $g\geq 6$ and $\delta\geq 3$ or that $g\geq 4$ and $\delta\geq 5$. Furthermore, we show that, for every positive integer $k$, there is a connected graph $G$ of girth at least $k$ and minimum degree at least $k$ such that $\alpha_b(G)\geq 2\left(1-\frac{1}{k}\right)\alpha(G)$. Our results imply that lower bounds on the girth and the minimum degree of a connected graph $G$ can lower the fraction $\frac{\alpha_b(G)}{\alpha(G)}$ from $4$ below $2$, but not any further.

Keywords

Main Subjects

#### References

[1] M. Ahmane, I. Bouchemakh, and E. Sopena, On the broadcast independence number of caterpillars, Discrete Appl. Math. 244 (2018), 20–35.
[2] S. Bessy and D. Rautenbach, Algorithmic aspects of broadcast independence, arXiv 1809.07248.
[3] S. Bessy and D. Rautenbach, Relating broadcast independence and independence, Manuscript 2018.
[4] I. Bouchemakh and M. Zemir, On the broadcast independence number of grid graph, Graphs Combin. 30 (2014), no. 1, 83–100.
[5] R. Diestel, Graduate Texts in Mathematics, Springer-Verlag New York, Incorporated, 2000.
[6] J.E. Dunbar, D.J. Erwin, T.W. Haynes, S.M. Hedetniemi, and S.T. Hedetniemi, Broadcasts in graphs, Discrete Appl. Math. 154 (2006), no. 1, 59–75.
[7] P. Erdős, Graph theory and probability II, Canad. J. Math. 13 (1961), 346–352.
[8] D.J. Erwin, Cost domination in graphs, (Ph.D. thesis), Western Michigan University, 2001.
[9] F. Joos and D. Rautenbach, Equality of distance packing numbers, Discrete Math. 338 (2015), no. 12, 2374–2377.
[10] J. Topp and L. Volkmann, On packing and covering numbers of graphs, Discrete Math. 96 (1991), no. 3, 229–238.