The minimum size of a $k$-connected locally nonforesty graph

TL;DR

The paper determines the exact minimum edge count $f(k,n)$ for $k$-connected locally nonforesty graphs on $n$ vertices. It shows that for $k\ge 5$ the local nonforesty constraint does not affect the minimum, yielding $f(k,n)=\lceil kn/2\rceil$, and focuses on the nontrivial cases $k\le 4$. For $k=4$ the minimum is $h(n)$ with a congruence-dependent form; for $k=2$ it is $g(n)$ with a piecewise dependence on $n\bmod 4$; and for connected graphs the minimum is $p(n)$ with another piecewise form. The extremal bounds are matched by constructions built from disjoint copies of $K_4$ linked in cycle- or path-like fashions, together with residue-adjusting replacements, and the analysis employs local-subgraph structure and block-cutpoint decompositions to derive tight lower bounds. These results elucidate how the locally nonforesty condition interacts with connectivity in extremal graph theory and provide exact thresholds for the minimum number of edges in low-connectivity regimes.

Abstract

A local subgraph of a graph is the subgraph induced by the neighborhood of a vertex. Thus a graph of order $n$ has $n$ local subgraphs. A graph $G$ is called locally nonforesty if every local subgraph of $G$ contains a cycle. Clearly, a graph is locally nonforesty if and only if every vertex of the graph is the hub of a wheel. We determine the minimum size of a $k$-connected locally nonforesty graph of order $n.$

Figures (2)

• Figure 1: $A_i,$$B_1,$$C_1$ and $D_1$
• Figure 2: $B_1,$$C_1,$$D_1$ and $D_2$