Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.01364

Maximizing the number of stars in graphs with forbidden properties

Zhanar Berikkyzy·Kirsten Hogenson·Rachel Kirsch·Jessica McDonald
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhanar Berikkyzy, Kirsten Hogenson, Rachel Kirsch, Jessica McDonald. 2025-04-02. Maximizing the number of stars in graphs with forbidden properties. https://arxiv.org/abs/2504.01364

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗