Maxima of the Q-index: Forbidden rainbow Hamilton paths, matchings and linear forests

Abstract

A classical result in graph theory, proved by Ore and Bondy independently, states that every graph on $n\ge 5$ vertices with more than $\left(\begin{array}{c}n-1\\ 2\end{array}\right)$ edges contains a path of length $n-1$. Another well-known result due to Erdős and Gallai asserts that for $k\ge 2$ and $n\ge 2k+2$, every n-vertex graph with more than $\max \{\left(\begin{array}{c}2k+1\\ 2\end{array}\right),\left(\begin{array}{c}k\\ 2\end{array}\right)+k(n-k\left)\right\}$ edges contains a matching of size $k+1$. In this paper, we establish several spectral analogues for graphs to admit a rainbow substructure. Let $G_1,G_2,\dots ,G_{n-1}$ be a collection of (not necessarily distinct) graphs on the same vertex set $\{1,2,\dots ,n\}$ and $q\left(G_i\right)$ be the spectral radius of the signless Laplacian matrix of $G_i$. Firstly, we prove that if $q\left(G_i\right)\ge 2(n-2)$ for every $i\in [n-1]$, then $G_1,G_2,\dots ,G_{n-1}$ admit a rainbow Hamilton path, i.e., a Hamilton path whose edges come from different $G_i$'s, unless $G_1=G_2=\cdots =G_{n-1}$ and $G_1\simeq K_{n-1}\cup I_1$. Secondly, we establish a sufficient condition on the signless Laplacian spectral radius of a family of graphs to guarantee a rainbow matching of size $k+1$. Thirdly, we study the spectral condition on graphs to admit a rainbow linear forests. The main ingredient of our proofs is the shifting techniques with its own interest, involving rainbow Hamilton paths, matchings and linear forests. These results present new examples and illustrations for which the rainbow Q-spectral problems have the adjacency spectral analogues. Finally, some related problems are proposed for further research.