On Betti numbers of edge ideals of crown graphs (Q1728774)

Let $G=(V, E)$ be a finite simple (no loops or multiple edges) undirected graph with vertex set $V(G)=\{x_1, x_2,\dots, x_n\}$ and edge set $E(G)$. Let $I(G)=(x_ix_j\mid \{x_i,x_j\}\in E(G))$ be an ideal in the polynomial ring $R=\mathbb{K}[x_1,\dots,x_n]$, where the vertex $x_i$ is identified with the variable $x_i$. The ideal $I(G)$ is called the edge ideal of $G$. \par The authors study edge ideals mainly to investigate relations between algebraic properties of edge ideals and combinatorial properties of graphs. They mainly focus on describing invariants of $I(G)$ in terms of $G$. The organization of the paper is as follows: In Sect. 2 the authors recall some definitions and introduce the notion of the strongly disjoint set of bouquets. Moreover, in this section, they recall some well-known results by \textit{M. Hochster} [in: Proc. Second Conf., Univ. Oklahoma, Norman, Okla. 171--223 (1975; Zbl 0351.13009)] and \textit{M. Katzman} [J. Comb. Theory, Ser. A 113, No. 3, 435--454 (2006; Zbl 1102.13024)]. In Sect. 3 the authors introduce the crown graphs and prove some results. In particular, they prove Theorems 3.1 and 3.9. In Sect. 4 the authors discuss domination parameters of graphs and present a result on projective dimension of edge ideals of crown graphs.