Sufficient conditions of $k$-leaf-connected graphs and spanning trees with bounded total $k$-excess

TL;DR

This paper establishes new sufficient conditions based on independence number and spectral properties for graphs to be $k$-leaf-connected and to contain spanning trees with bounded total $k$-excess, extending classical theorems.

Contribution

It proves a new bound on independence number ensuring $k$-leaf-connectedness and provides spectral criteria for spanning trees with limited $k$-excess.

Findings

Proves that $ ext{independence number} \, ext{α}(G) \, ext{≤} \, m-k+1$ guarantees $k$-leaf-connectedness in $m$-connected graphs.

Provides spectral conditions for the existence of spanning trees with total $k$-excess at most $b$.

Extends classical Hamilton-connectedness results to $k$-leaf-connected graphs.

Abstract

Chv\'{a}tal and Erd\"{o}s [Discrete Math. 2 (1972) 111-113] stated that, for an $m$-connected graph $G$, if its independence number $\alpha(G)\leq m-1$, then $G$ is Hamilton-connected. Note that $k$-leaf-connectedness is a natural generalization of Hamilton-connectedness of a graph. Ozeki and Yamashita [Graphs Combin. 27 (2011) 1-26] posed an open problem: What is the sufficient condition based on the independence number for an $m$-connected graph to be $k$-leaf-connected? In this paper, we prove that if $\alpha(G)\leq m-k+1,$ then an $m$-connected graph $G$ is $k$-leaf-connected. This not only answers the open problem of Ozeki and Yamashita, but also extends Chv\'{a}tal-Erd\"{o}s Theorem. As applications, we present sufficient spectral conditions for an $m$-connected graph to be $k$-leaf-connected. Let $k\geq 2$ be an integer and $T$ be a spanning tree of a connected graph. The total $k$-excess $te(T,k)$ is the summation of the $k$-excesses of all vertices in $T$, namely, $te(T,k)=\sum_{v\in V(T)}\mbox{max}\{0, d_{T}(v)-k\}.$ One can see that $T$ is a spanning $k$-tree if and only if $te(T,k)=0$. Fan, Goryainov, Huang and Lin [Linear Multilinear Algebra 70 (2022) 7264-7275] presented sufficient spectral conditions for a connected graph to contain a spanning $k$-tree. We in this paper propose sufficient conditions in terms of the spectral radius for a connected graph to contain a spanning tree with $te(T,k)\leq b$, where $b\geq0$ is an integer.