Erd\"os-Gallai-type results for conflict-free connection of graphs

TL;DR

This paper explores Erdős-Gallai-type conditions for the conflict-free connection number in graphs, establishing bounds and criteria for when graphs can be made conflict-free connected with a limited number of colors.

Contribution

It introduces Erdős-Gallai-type theorems specifically for the conflict-free connection number, providing new theoretical bounds and conditions for graph coloring.

Findings

Derived bounds for conflict-free connection numbers

Established criteria for conflict-free connectivity

Extended Erdős-Gallai results to conflict-free graph coloring

Abstract

A path in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges. An edge-colored graph is called \emph{conflict-free connected} if there is a conflict-free path between each pair of distinct vertices. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $\mathit{cfc}(G)$, is defined as the smallest number of colors that are required to make $G$ conflict-free connected. In this paper, we obtain Erd\"{o}s-Gallai-type results for the conflict-free connection numbers of graphs.

Equations

$$g(n,k)=\left\{\begin{array}[]{rcl}{n\choose 2}-1,&&{k=2}\\ n-1,&&{3\leq k<\lceil\log_{2}n\rceil}\\ does\ not\ exist,&&{\lceil\log_{2}n\rceil\leq k\leq n-1.}\end{array}\right.$$

Full text

Keywords: conflict-free connection coloring; conflict-free connection number; Erdös-Gallai-type result.

AMS subject classification 2010: 05C15, 05C40, 05C35.

1 Introduction

All graphs mentioned in this paper are simple, undirected and finite. We follow book [1] for undefined notation and terminology. Let $P_{1}=v_{1}v_{2}\cdots v_{s}$ and $P_{2}=v_{s}v_{s+1}\cdots v_{s+t}$ be two paths. We denote $P=v_{1}v_{2}\cdots v_{s}v_{s+1}\cdots v_{s+t}$ by $P_{1}\odot P_{2}$. Coloring problems are important subjects in graph theory. The hypergraph version of conflict-free coloring was first introduced by Even et al. in [7]. A hypergraph $H$ is a pair $H=(X,E)$ where $X$ is the set of vertices, and $E$ is the set of nonempty subsets of $X$, called hyper-edges. The conflict-free coloring of hypergraphs was motivated to solve the problem of assigning frequencies to different base stations in cellular networks, which is defined as a vertex coloring of $H$ such that every hyper-edge contains a vertex with a unique color.

Later on, Czap et al. in [6] introduced the concept of conflict-free connection colorings of graphs motivated by the conflict-free colorings of hypergraphs. A path in an edge-colored graph $G$ is called a conflict-free path if there is a color appearing only once on the path. The graph $G$ is called conflict-free connected if there is a conflict-free path between each pair of distinct vertices of $G$. The minimum number of colors required to make a connected graph $G$ conflict-free connected is called the conflict-free connection number of $G$, denoted by $\mathit{cfc}(G)$. If one wants to see more results, the reader can refer to [3, 4, 5, 6]. For a general connected graph $G$ of order $n$, the conflict-free connection number of $G$ has the bounds $1\leq\mathit{cfc}(G)\leq n-1$. When equality holds, $\mathit{cfc}(G)=1$ if and only if $G=K_{n}$ and $\mathit{cfc}(G)=n-1$ if and only if $\mathit{cfc}(G)=K_{1,n-1}$.

The Erdös-Gallai-type problem is an interesting problem in extremal graph theory, which was studied in [9, 10, 11, 12] for rainbow connection number $\mathit{rc}(G)$; in [8] for proper connection number $\mathit{pc}(G)$; in [2] for monochromatic connection number $\mathit{mc}(G)$. We will study the Erdös-Gallai-type problem for the conflict-free number $\mathit{cfc}(G)$ in this paper.

2 Auxiliary results

At first, we need some preliminary results.

Lemma 2.1

[6] Let $u,v$ be distinct vertices and let $e=xy$ be an edge of a 2-connected graph. Then there is a $u-v$ path in $G$ containing the edge $e$.

For a 2-edge connected graph, the authors [5] presented the following result:

Theorem 2.2

[5] If $G$ is a $2-$edge connected graph, then $\mathit{cfc}(G)=2$.

For a tree $T$, there is a sharp lower bound:

Theorem 2.3

[4] Let $T$ be a tree of order $n$. Then $\mathit{cfc}(T)\geq\mathit{cfc}(P_{n})=\lceil\log_{2}n\rceil$.

Lemma 2.4

Let $G$ be a connected graph and $H=G-B$, where $B$ denotes the set of the cut-edges of $G$. Then $\mathit{cfc}(G)\leq\max\{2,|B|\}$.

Proof. If $B$=$\varnothing$, then by Theorem 2.2, $\mathit{cfc}(G)$=2. If $|B|\geq 1$, then all the blocks are non-trivial in each component of $G-B$. Now we give $G$ a conflict-free coloring: assign one edge with color 1 and the remaining edges with color 2 in each block of each component of $G-B$; for the edges $e\in B$, we assign each edge with a distinct color from $\{1,2,\cdots,|B|\}$.

Now we check every pair of vertices. Let $u$ and $v$ be arbitrary two vertices. Consider first the case that $u$ and $v$ are in the same component of $G-B$. If $u$ and $v$ are in the same block, by Lemma 2.1 there is a conflict-free $u-v$ path. If $u,v$ are in different blocks, let $P=P_{1}\odot P_{2}\odot\cdots\odot P_{r}$ be a $u-v$ path, where $P_{i}$ $(i\in[r])$ is the path in each block of the component. Then we can choose a conflict-free path in one block, say $P_{1}$, and choose a monochromatic path with color 2 in each block of the remaining blocks, say $P_{i}$ $(2\leq i\leq r-1)$, clearly, $P$ is a conflict-free $u-v$ path. Now consider the case that $u$ and $v$ are in distinct components of $G-B$. If there exists one cut-edge $e$ with color $c\notin\{1,2\}$, then there is a conflict-free $u-v$ path since the color used on $e$ is unique. If there does not exist cut-edge with color $c\notin\{1,2\}$, then suppose that there is only one cut-edge $e=xy$ with color 1, without loss of generality, let $u,x$ be in a same component and $v,y$ be in a same component. We choose a monochromatic $u-x$ path $P_{1}$ with color 2 and choose a monochromatic $v-y$ path $P_{2}$ with 2, then $P=P_{1}xyP_{2}$ is a conflict-free $u-v$ path. If there is only one cut-edge $e=st$ colored by 2, without loss of generality, then we say $u,s$ are in the same component and $t,v$ in a same component, we choose a monochromatic $u-s$ path $P_{1}$ and a conflict-free $t-v$ path $P_{2}$ in each component. Then $P=P_{1}stP_{2}$ is a conflict-free $u-v$ path. If there are exactly two cut-edges $e_{1}=st$ and $e_{2}=xy$ colored by 1 and 2, respectively, without loss of generality, we say that $u,s$ are in a same component, $t,x$ are in a same component and $y,v$ are in a same component. Then we choose a monochromatic $u,s$ path $P_{1}$, $t,x$ path $P_{2}$ and $y,v$ path $P_{3}$ in the three components, respectively, with color 2. Hence, $P=P_{1}stP_{2}xyP_{3}$ is a conflict-free $u-v$ path. So, we have $\mathit{cfc}(G)\leq\max\{2,|B|\}$. $\Box$

Lemma 2.5

Let $G$ be a connected graph of order $n$ with $k$ cut-edges. Then

$|E(G)|\leq{n\choose k}+k$**

.

Proof. Clearly, it holds for $k=0$. Assuming that $k\geq 1$. Let $G$ be a maximal graphs with $k$ cut-edges. Let $B$ be the set of all the bridges. And let $G-B$ be the graph by deleting all the cut-edges. Let $C_{1},C_{2},\cdots,C_{k+1}$ be the components of $G-B$ and $n_{i}$ be the orders of $C_{i}$. Then E(G)=\sum_{i=1}^{k+1}$$n_{i}\choose 2$$+k. Let $C_{i}$ and $C_{j}$ be two components of $G-B$ with $1<n_{i}\leq n_{j}$. Now we construct a graph $G^{\prime}$ by moving a vertex $v$ from $C_{i}$ to $C_{j}$, replace $v$ with an arbitrary vertex in $V(C_{k})\setminus v$ for the cut-edges incident with $v$, add the edges between $v$ and the vertices in $C_{j}$, and delete the edges between $v$ and the vertices in $C_{i}$, where $v$ is not adjacent to the vertices of $C_{i}$. Now we have |E(G^{\prime})|=\sum_{s=1\neq i,j}^{k+1}$$n_{s}\choose 2+$n_{i}-1\choose 2$+$n_{j}+1\choose 2$+$k$=\sum_{s=1\neq i,j}^{k+1}$$n_{s}\choose 2+$n_{i}\choose 2$-$n_{i}-1$+$n_{j}\choose 2$+$n_{j}+k$=|E(G)|+n_{j}-n_{i}+1$$>|E(G)|. When we do repetitively the operation, we have $|E(G)|\leq{n\choose k}+k$. $\Box$

3 Main results

Now we consider the Erdös-Gallai-type problems for $\mathit{cfc}(G)$. There are two types, see below.

Problem 3.1

For each integer $k$ with $2\leq k\leq n-1$, compute and minimize the function $f(n,k)$ with the following property: for each connected graph $G$ of order $n$, if $|E(G)|\geq f(n,k)$, then $\mathit{cfc}(G)\leq k$.

Problem 3.2

For each integer $k$ with $2\leq k\leq n-1$, compute and maximize the function $g(n,k)$ with the following property: for each connected graph $G$ of order $n$, if $|E(G)|\leq g(n,k)$, then $\mathit{cfc}(G)\geq k$.

Clearly, there are two parameters which are equivalent to $f(n,k)$ and $g(n,k)$ respectively. For each integer $k$ with $2\leq k\leq n-1$, let $s(n,k)=\max\{|E(G)|:|V(G)|=n,\mathit{cfc}\geq k\}$ and $t(n,k)=\min\{|E(G)|:|V(G)|=n,\mathit{cfc}\leq k\}$. By the definitions, we have $g(n,k)=t(n,k-1)-1$ and $f(n,k)=s(n,k+1)+1$.

Using Lemma 2.4 we first solve Problem 3.1.

Theorem 3.3

f(n,k)=$$n-k-1\choose 2$$+k+2* for $2\leq k\leq n-1$.*

Proof. At first, we show the following claims.

Claim 1: For $k\geq 2$, f(n,k)\leq$$n-k-1\choose 2$$+k+2.

Proof of Claim 1: We need to prove that for any connected graph $G$, if E(G)\geq$$n-k-1\choose 2$$+k+2, then $\mathit{cfc}(G)\leq k$. Suppose to the contrary that $\mathit{cfc}(G)\geq k+1$. By Lemma 2.4, we have $|B|\geq k+1$. By Lemma 2.5, E(G)\leq$$n-k-1\choose 2$$+k+1, which is a contradiction.

Claim 2: For $k\geq 2$, f(n,k)\geq$$n-k-1\choose 2$$+k+2.

Proof of Claim 2: We construct a graph $G_{k}$ by identifying the center vertex of a star $S_{k+2}$ with an arbitrary vertex of $K_{n-k-1}$. Clearly, E(G_{k})=$$n-k-1\choose 2$$+k+1. Since $\mathit{cfc}(S_{k+2})=k+1$, then $\mathit{cfc}(G_{k})\geq k+1$. It is easy to see that $\mathit{cfc}(G_{k})=k+1$. Hence, f(n,k)\geq$$n-k-1\choose 2$$+k+2.

The conclusion holds from Claims 1 and 2. $\Box$

Now we come to the solution for Problem 3.2, which is divided as three cases.

Lemma 3.4

For $k=2$, $g(n,2)$=n\choose 2$$-1.

Proof. Let $G$ be a complete graph of order $n$. The number of edges in $G$ is $n\choose 2$, $i.e.$, E(G)=$$n\choose 2. Clearly, when g(n,2)=$$n\choose 2$$-1 for every $G$, $\mathit{cfc}(G)\geq 2$. $\Box$

Lemma 3.5

For every integer $k$ with $3\leq k<\lceil\log_{2}n\rceil$, $g(n,k)=n-1$.

Proof. We first give an upper bound of $t(n,k)$. Let $C_{n}$ be a cycle. Then $t(n,k)\leq n$ since $\mathit{cfc}(C_{n})=2\leq k$. And then, we prove that $t(n,k)=n$. Suppose $t(n,k)\leq n-1$. Let $P_{n}$ be a path with size $n-1$. Since $\mathit{cfc}(P_{n})=\lceil\log_{2}n\rceil$ by Theorem 2.3, it contradicts the condition the $k<\lceil\log_{2}n\rceil$. So $t(n,k)=n$. By the relation that $g(n,k)=t(n,k-1)-1$, we have $g(n,k)=n-1$. $\Box$

Lemma 3.6

For $k\geq\lceil\log_{2}n\rceil$, $g(n,k)$ does not exist.

Proof. Let $P_{n}$ be a path. Then we have $t(n,k)\leq n-1$ since $\mathit{cfc}(P_{n})=\lceil\log_{2}n\rceil$. And since $t(n,k)\geq n-1$, it is clear that $t(n,k)=n-1$. Since every graph $G$ is connected, $g(n,k)\geq n-1$. By the relation that $g(n,k)=t(n,k-1)-1$, we have $g(n,k)=n-2$ for $k\geq\lceil\log_{2}n\rceil$, which contradicts the connectivity of graphs. $\Box$

Combining Lemmas 3.4, 3.5 and 3.6, we get the solution for Problem 3.2.

Theorem 3.7

For $k$ with $2\leq k\leq n-1$,

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