The maximum number of triangles in graphs without large linear forests

TL;DR

This paper determines upper bounds on the number of edges and triangles in graphs that do not contain large linear forests, extending previous results by incorporating minimum degree constraints.

Contribution

It generalizes existing bounds on edges and triangles in graphs without large linear forests by including minimum degree conditions.

Findings

Established new upper bounds on edges for graphs with given linear forest size and minimum degree.

Derived maximum number of triangles in such graphs, depending on parity of parameters.

Extended previous results to more general graph classes with degree constraints.

Abstract

Let $G$ be a graph on $n$ vertices. A linear forest is a graph consisting of vertex-disjoint paths and isolated vertices. A maximum linear forest of $G$ is a subgraph of $G$ with maximum number of edges, which is a linear forest. We denote by $l(G)$ this maximum number. Let $t=\left\lfloor (k-1)/2\right \rfloor$. Recently, Ning and Wang \cite{boning} proved that if $l(G)=k-1$, then for any $k<n$ \[ e(G) \leq \max \left\{\binom{k}{2},\binom{t}{2}+t (n - t)+ c \right\}, \] where $c=0$ if $k$ is odd and $c=1$ otherwise, and the inequality is tight. In this paper, we prove that if $l(G)=k-1$ and $\delta(G)=\delta$ ($\delta<\lfloor k/2 \rfloor$), then for any $k<n$ \[ e(G) \leq \max \left\{\binom{k-\delta}{2}+\delta(n-k+\delta),\binom{t}{2}+t\left(n-t\right)+c \right\}. \] When $\delta=0$, it reduces to Ning and Wang's result. Moreover, let $r_3(G)$ be the number of triangles in $G$. We prove that if $l(G)=k-1$ and $\delta(G)= \delta$, then for any $k<n$ \[ r_3(G)\leq \max \left\{\binom{k-\delta}{3}+\binom{\delta}{2}(n-k+\delta),\binom{t}{3}+\binom{t}{2}\left(n-t\right)+d \right\}. \] where $d=0$ if $k$ is odd and $d=t$ otherwise.