Strongly light subgraphs in the 1-planar graphs with minimum degree 7

TL;DR

This paper investigates the existence of specific small subgraphs within 1-planar graphs that have a minimum degree of 7, demonstrating the presence of strongly light subgraphs with bounded degrees.

Contribution

It identifies certain strongly light subgraphs in 1-planar graphs with minimum degree 7, expanding understanding of their structural properties.

Findings

Existence of strongly light subgraphs in 1-planar graphs with minimum degree 7

Bounded degree subgraphs are guaranteed in these graphs

Contributes to the structural theory of 1-planar graphs

Abstract

A graph is {\em $1$-planar} if it can be drawn in the plane such that every edge crosses at most one other edge. A connected graph $H$ is {\em strongly light} in a family of graphs $\mathfrak{G}$, if there exists a constant $\lambda$, such that every graph $G$ in $\mathfrak{G}$ contains a subgraph $K$ isomorphic to $H$ with $\deg_{G}(v) \leq \lambda$ for all $v \in V(K)$. In this paper, we present some strongly light subgraphs in the family of $1$-planar graphs with minimum degree~$7$.