Maximizing the number of stars in graphs with forbidden properties

TL;DR

This work generalizes extremal-graph results from maximizing edges in not-quite-Hamiltonian graphs to maximizing $t$-stars, across several Hamiltonian-type properties, by introducing the generalized endpoint families $G_n^{\ell}(i)$ and their spanning-subgraph variants. A concavity-based argument shows that, under a minimum-degree constraint, the maximum $t$-star count is achieved at the endpoint graphs $G_n^{\ell}(i_d)$ or $G_n^{\ell}(i_0)$, with a precise extremal-graph description depending on whether these two endpoints yield equal or unequal counts; a detailed characterization is given via the families $\mathcal{G}^{\ell}_n(i)$. The paper also handles the $k$-connected setting using the $H_n^{k}(i)$ family and provides partial progress on removing the minimum-degree condition, along with a separate analysis for the maximum number of $t$-stars in non-$k$-connected graphs. These results extend classical Erdős and Füredi–Kostochka–Luo theorems to a broader class of forbidden properties and to $t$-star counts, offering a unified extremal framework and precise endpoint behavior across regimes of $t$ and $\ell$.

Abstract

Erdős proved an upper bound on the number of edges in an $n$-vertex non-Hamiltonian graph with given minimum degree and showed sharpness via two members of a particular graph family. Füredi, Kostochka and Luo showed that these two graphs play the same role when ``number of edges'' is replaced by ``number of t-stars,'' and that two members of a more general graph family maximize the number of edges among non-$k$-edge-Hamiltonian graphs. In this paper we generalize their former result from Hamiltonicity to related properties (traceability, Hamiltonian-connectedness, $k$-edge Hamiltonicity, $k$-Hamiltonicity) and their latter result from edges to $t$-stars. We identify a family of extremal graphs for each property that is forbidden. This problem without the minimum degree condition was also open; here we conjecture a complete description of the extremal family for each property, and prove the characterization in some cases. Finally, using a different family of extremal graphs, we find the maximum number of $t$-stars in non-$k$-connected graphs.

Theorems & Definitions (40)

• Theorem 1: Erdős ErdosRemarks
• Theorem 2: Füredi, Kostochka, Luo FKL18
• Theorem 3: Füredi, Kostochka, Luo FKL19
• Proposition 3
• Theorem 4
• Theorem 5
• Theorem 6
• Corollary 7
• Theorem 8
• Conjecture 9