Independent dominating sets in graphs of girth five

<jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548320000279_inline1.png" /><jats:tex-math> $\gamma(G)$ </jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548320000279_inline2.png" /><jats:tex-math> $${\gamma _ \circ }(G)$$ </jats:tex-math></jats:alternatives></jats:inline-formula> denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if <jats:italic>G</jats:italic> is an <jats:italic>n</jats:italic>-vertex graph of minimum degree at least <jats:italic>d</jats:italic>, then<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" position="float" xlink:href="S0963548320000279_eqnu1.png" /><jats:tex-math>$$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>In this paper the main result is that if <jats:italic>G</jats:italic> is any <jats:italic>n</jats:italic>-vertex <jats:italic>d</jats:italic>-regular graph of girth at least five, then<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" position="float" xlink:href="S0963548320000279_eqnu2.png" /><jats:tex-math>$$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$</jats:tex-math></jats:alternatives></jats:disp-formula>for some constant <jats:italic>c</jats:italic> independent of <jats:italic>d</jats:italic>. This result is sharp in the sense that as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548320000279_inline3.png" /><jats:tex-math> $d \rightarrow \infty$ </jats:tex-math></jats:alternatives></jats:inline-formula>, almost all <jats:italic>d</jats:italic>-regular <jats:italic>n</jats:italic>-vertex graphs G of girth at least five have<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" position="float" xlink:href="S0963548320000279_eqnu3.png" /><jats:tex-math>$$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>Furthermore, if <jats:italic>G</jats:italic> is a disjoint union of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548320000279_inline4.png" /><jats:tex-math> ${n}/{(2d)}$ </jats:tex-math></jats:alternatives></jats:inline-formula> complete bipartite graphs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548320000279_inline5.png" /><jats:tex-math> $K_{d,d}$ </jats:tex-math></jats:alternatives></jats:inline-formula>, then <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548320000279_inline6.png" /><jats:tex-math> ${\gamma_\circ}(G) = \frac{n}{2}$ </jats:tex-math></jats:alternatives></jats:inline-formula>. We also prove that there are <jats:italic>n</jats:italic>-vertex graphs G of minimum degree <jats:italic>d</jats:italic> and whose maximum degree grows not much faster than <jats:italic>d</jats:italic> log <jats:italic>d</jats:italic> such that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548320000279_inline7.png" /><jats:tex-math> ${\gamma_\circ}(G) \sim {n}/{2}$ </jats:tex-math></jats:alternatives></jats:inline-formula> as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548320000279_inline8.png" /><jats:tex-math> $d \rightarrow \infty$ </jats:tex-math></jats:alternatives></jats:inline-formula>. Therefore both the girth and regularity conditions are required for the main result.</jats:p>

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