[Paper Review] Proper connection numbers of complementary graphs

Fei Huang, Xueliang Li+1 more|arXiv (Cornell University)|Apr 9, 2015

Advanced Graph Theory Research9 references4 citations

TL;DR

This paper investigates the proper connection number of a graph's complement, establishing a Nordhaus-Gaddum-type inequality: for connected complementary graphs $G$ and $¯{G}$, the sum $pc(G) + pc(\overline{G})$ satisfies $4 \leq pc(G) + pc(\overline{G}) \leq n$, with equality in the upper bound only for trees with maximum degree $n-2$. The work characterizes graphs with proper connection number $n-2$ and uses this to derive the tight bound on the sum of connection numbers.

ABSTRACT

A path $P$ in an edge-colored graph $G$ is called a proper path if no two adjacent edges of $P$ are colored the same, and $G$ is proper connected if every two vertices of $G$ are connected by a proper path in $G$. The proper connection number of a connected graph $G$, denoted by $pc(G)$, is the minimum number of colors that are needed to make $G$ proper connected. In this paper, we investigate the proper connection number of the complement of graph $G$ according to some constraints of $G$ itself. Also, we characterize the graphs on $n$ vertices that have proper connection number $n-2$. Using this result, we give a Nordhaus-Gaddum-type theorem for the proper connection number. We prove that if $G$ and $\overline{G}$ are both connected, then $4\le pc(G)+pc(\overline{G})\le n$, and the only graph attaining the upper bound is the tree with maximum degree $Δ=n-2$.

Motivation & Objective

• To investigate the proper connection number of the complement of a graph $G$ under structural constraints on $G$.
• To characterize all $n$-vertex graphs with proper connection number $n-2$.
• To establish a tight Nordhaus-Gaddum-type inequality for the proper connection number of complementary graphs.
• To determine the extremal graphs achieving the upper bound in the sum of proper connection numbers of $G$ and $\overline{G}$.

Proposed method

• Characterize graphs with proper connection number $n-2$ by analyzing their structure and edge-coloring properties.
• Use known results on proper-path colorings and spanning subgraphs to bound $pc(G)$ and $pc(\overline{G})$.
• Apply the strong property of proper-path colorings to ensure connectivity with minimal colors.
• Analyze the complement graph $\overline{G}$ under different structural cases: no cut vertices, cut vertices with multiple components, and cut vertices with two components.
• Use case analysis based on the number of components in $G - u$ for cut vertices $u$, and apply results on complete $k$-partite subgraphs in $\overline{G}$.
• Leverage known bounds on $pc(H)$ for 2-connected and tree subgraphs to derive upper bounds on $pc(G)$ and $pc(\overline{G})$.

Experimental results

Research questions

• RQ1 What is the maximum possible value of $pc(G) + pc(\overline{G})$ for connected complementary graphs $G$ and $\overline{G}$?
• RQ2 Which graphs on $n$ vertices achieve the proper connection number $n-2$?
• RQ3 Under what conditions does $pc(G) + pc(\overline{G}) = n$ hold for connected $G$ and $\overline{G}$?
• RQ4 How does the presence of cut vertices in $G$ affect the proper connection number of $\overline{G}$?
• RQ5 Can a Nordhaus-Gaddum-type inequality be established for the proper connection number, and if so, what is the tightest possible bound?

Key findings

• The sum $pc(G) + pc(\overline{G})$ is bounded below by 4 and above by $n$ for any connected complementary graphs $G$ and $\overline{G}$.
• The upper bound $pc(G) + pc(\overline{G}) = n$ is attained if and only if $G$ is a tree with maximum degree $n-2$.
• For $n \geq 7$, $pc(G) + pc(\overline{G}) < n$ unless $G$ is a tree with $\Delta = n-2$, which achieves equality.
• If $G$ has a cut vertex $u$ such that $G - u$ has at least three components, then $pc(G) + pc(\overline{G}) < n$.
• When $G$ has cut vertices each incident with only one pendant edge, $pc(G) \leq 3$ and $pc(\overline{G}) \leq \max\{3, n-4\}$, leading to $pc(G) + pc(\overline{G}) < n$ for $n \geq 7$, and $< n$ for $n=6$.
• The characterization of graphs with $pc(G) = n-2$ is essential in proving the tight upper bound and identifying the extremal case.