Abstract
In the 1970s, Erdős asked how many edges are needed in a graph on $n$ vertices, to ensure the existence of a cycle of length exactly $n-k$. In this paper, we consider the spectral analog of Erdős' problem. Indeed, the problem of determining tight spectral radius conditions for cycles of length $\ell$ in a graph of order $n$ for each $\ell \in[3,n]$ seems very difficult. We determine tight spectral radius conditions for $C_{\ell}$ where $\ell$ belongs to an interval of the form $[n-\Theta(\sqrt{n}),n]$. As a main tool, we prove a stability result of a theorem due to Woodall, which states that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is an integer, if $e(G)>\binom{n-k-1}{2}+\binom{k+2}{2}$ then $G$ contains a $C_{\ell}$ for each $\ell\in [3,n-k]$. We prove a tight spectral condition for the circumference of a $2$-connected graph with a given minimum degree, of which the main tool is a stability version of a 1976 conjecture of Woodall on circumference of a $2$-connected graph with a given minimum degree proved by Ma and the second author. We also give a brief survey on this area and point out where we are and our predicament.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics