The Path Partition Conjecture is True and its Validity Yields Upper Bounds for Detour Chromatic Number and Star Chromatic Number

The Path Partition Conjecture is True and its Validity Yields Upper Bounds for Detour Chromatic Number and Star Chromatic Number

Abstract

The detour order of a graph , denoted , is the order of a longest path in . A partition of such that and is called an -partition of . A graph is called -partitionable if has an -partition for every pair of positive integers such that . The well-known Path Partition Conjecture states that every graph is -partitionable. In [7] Dunber and Frick have shown that if every 2-connected graph is -partitionable then every graph is -partitionable. In this paper we show that every 2-connected graph is -partitionable. Thus, our result settles the Path Partition Conjecture affirmatively. We prove the following two theorems as the implications of the validity of the Path Partition Conjecture.
Theorem 1: For every graph , , where is the star chromatic number of a graph .

The detour chromatic number of a graph , denoted , is the minimum number of colours required for colouring the vertices of such that no path of order greater than is mono coloured. These chromatic numbers were introduced by Chartrand, Gellar and Hedetniemi[5] as a generalization of vertex chromatic number .
Theorem 2: For every graph and for every , , where denote the detour chromatic number.
Theorem 2 settles the conjecture of Frick and Bullock [9] that , for every graph , for every , affirmatively.

Keywords:Path Partition;Path Partition Conjecture;Star Chromatic Number;Detour Chromatic Number;Upper bound of chromatic number;Upper bound of Star Chromatic Number;Upper bound of Detour Chromatic Number.

1 Introduction

All graphs considered here are simple, finite and undirected. Terms not defined here can be referred from the book [20]. A longest path in a graph is called a detour of . The number of vertices in a detour of is called the detour order of and is denoted by . A partition of such that and is called an -partition of . If has an -partition for every pair of positive integers such that , then we say that is -partitionable. The following conjecture is popularly known as the Path Partition Conjecture.

Path Partition Conjecture: Every graph is -partitionable.

The Path Partition Conjecture was discussed by Lovasz and Mihok in 1981 in Szeged and treated in the theses [13] and [19]. The Path Partition Conjecture first appeared in the literature in 1983, in a paper by Laborde et al. [14]. In 1995 Bondy [2] posed the directed version of the Path Partition Conjecture. In 2004, Aldred and Thomassen [1] disproved two stronger versions of the Path Partition Conjecture, known as the Path Kernel Conjecture [4, 16] and the Maximum -free Set Conjecture [8]. Similar partitions were studied for other graph parameters too. Lovasz proved in [15] that every graph is -partitionable, where denotes the maximum degree (A graph is -partitionable if, for every pair of positive integers satisfying , there exists a partition of such that and ). For the results pertaining to the Path Partition Conjecture and related conjectures refer [3, 4, 6, 7, 8, 9, 10, 13, 14, 16, 18, 19, 17]. An -detour colouring of a graph is a colouring of the vertices of such that no path of order greater than is monocoloured. The detour chromatic number of graph , denoted by , is the minimum number of colours required for an -detour colouring of a graph . It is interesting to note that for a graph , when , . These chromatic numbers were introduced by Chartrand, Gellor and Hedetnimi [5] in 1968 as a generalization of vertex chromatic number.

If the Path Partition Conjecture is true, then the following conjecture of Frick and Bullock [9] is also true.

Frick-Bullock Conjecture: for every graph and for every .
Recently, Dunbar and Frick [7] proved the following theorem.

Theorem 1.1 (Dunber and Frick [7]).

If every 2-connected graph is -partitionable then every graph is -partitionable.

In this paper we show that the Path Partition Conjecture is true for every 2-connected graph. Thus, Theorem 1.1 and our result imply that the Path Partition Conjecture is true. The validity of the Path Partition Conjecture would imply the following Path Partition Theorem.

Path Partition Theorem. For every graph and for every -tuple of positive integers with and , there exists a partition of such that , for every , .
The Path Partition Theorem immediately implies that the Conjecture of Frick and Bullock is true. The validity of Frick and Bullock Conjecture naturally implies the classical upper bound for the chromatic number of a graph that proved by Gallai[11].

A star colouring of a graph is a proper vertex colouring in which every path on four vertices uses at least three distinct colours. The star chromatic number of denoted by is the least number of colours needed to star color . As a consequence of the Path Partition Theorem, we have obtained an upper bound for the star chromatic number. More precisely, we show that for every graph .

2 Main Result

In this section we prove our main result that every 2-connected graph is -partitionable.

We use Whitney’s Theorem on the characterization of 2-connected graph in the proof of our main result given in Theorem 2.2.

An ear of a graph is a maximal path whose internal vertices have degree 2 in . An ear decomposition of is a decomposition such that is a cycle and for is an ear of .

Theorem 2.1 (Whitney [21]).

A graph is 2-connected if and only if it has an ear decomposition. Furthermore, every cycle in a 2-connected graph is the initial cycle in some ear decomposition.

Theorem 2.2.

Every 2-connected graph is -partitionable.

Proof.

Let be a 2-connected graph. By Whitney’s Theorem there exists an ear decomposition , where is a cycle and for is an ear of . We prove that is -partitionable by induction on .

When , . Then . Thus, is a cycle. As every cycle is -partitionable, is -partitionable. By induction, we assume that if is any 2-connected graph having an ear decomposition , that is, with , then is -partitionable.

Let be a 2-connected graph with an ear decomposition . That is, . We claim that is
-partitionable. Let be a pair of positive integers with . Since is having the ear decomposition , can be considered as a 2-connected graph obtained from the 2-connected graph having the ear decomposition by adding a new path (ear) to , where and are new vertices to . As is a 2-connected graph having the ear decomposition with , by induction is -partitionable. Let be a pair of positive integers such that , with . Since is -partitionable, there exists an partition of such that and . In order to prove our claim that is -partitionable, we define an -partition of from the partition of as well as using the path . The construction of an -partition of is given under three cases, depending on , and , where is the number of new vertices in the path .

Case 1.

Then , where and are the vertices of .
Thus, . This implies, .

Case 1.1. Suppose and are in different parts of the partition of

.

Then, as and are in different parts of the partition of , the introduction of the new edge between the vertices and does not increase the length of any path either in or in . Further, as , we have Thus, is a required -partition of . Case 1.2. Suppose and are in the same part of the partition of

.

Without loss of generality, we assume that and are in .
Suppose . Then, as , the is a required -partition of .
Suppose , then observe that the addition of the edge to has increased the order of some of the longest paths (at least one longest path) in from to , where . On the other hand, any path of order in must contain the edge also.

Let be any path of order . Then, note that the edge for some , and .

Observation 2.1.

If we remove the vertex from the path , then we obtain two subpaths , say and , say of . The number of vertices in is exactly and the number of vertices in is .

Observation 2.2.

Consider the subpath of . Then observe that the end vertex of cannot be adjacent to any of the end vertices of any path of order in the induced subgraph in .

For, suppose is adjacent to an end vertex of a path, say of order in . Let . Without loss of generality, let be adjacent to . Then, there exists a path of order , a contradiction (Similar contradiction hold good if is adjacent ).

Let be the set of all paths in of order at least . For , let denote the terminus vertex of the subpath of of order and having its origin as the origin of . Let be the set of distinct vertices from the vertices , where . Suppose induces any path in . Consider any such path , where . Then, for , any vertex divides the path into three subpaths , , and .

Figure 1: Structures of various paths of order in

Claim 1. For every , , the vertex cannot be adjacent to any of the end vertices of any path of order greater than or equal to in , where or .
First we ascertain in Observation 2.3 then we prove the Claim 1.

Observation 2.3.

For, suppose . If , then consider the path,

in having vertices. As , the path has at least vertices. This implies there exists a path of order at least in . A contradiction to the fact that . Similarly, if , then consider the path,

in having vertices. As , the path has at least vertices. This implies that there exists a path of order at least in in . A contradiction to the fact that . Hence, .

To prove Claim 1, we suppose , for some , is adjacent to an end vertex of a path of order in . Let be a path of order in such that (without loss of generality) is adjacent to the vertex .

Case 1.2a.
Then consider the path , where is a subpath of of order having the vertex , the origin of as its origin. As is the path in such that is adjacent to , it follows that is a path in having the order , a contradiction.

Case 1.2b.

Then consider the path , where is the subpath of of order having the vertex , the origin of as its origin. As is the path in such that is adjacent to , it follows that is a path in having the order , a contradiction.
Hence the Claim 1.

Thus, it follows from the Claim 1 that

(1)

From Observation 1, it follows that

(2)

Let and . Then, from (1) and (2) it follows that and .
Hence is a required -partition of .

Case 2.
Then .

Case 2.1. Both and belong to the same partition or .
Without loss of generality, we assume that . That is, . Then is a required -partition of .
Case 2.2. The vertices and belong to different partitions and .

Without loss of generality, we assume that and . If is not an end vertex of a path of order in , then is a required -partition of . If is an end vertex of a path of order in , then cannot be an end-vertex of a path of order in (otherwise, would have a path of order ). Therefore is a required -partition of .

Case 3.
Colour all vertices of with red colour and colour all the vertices of with blue colour. Since the vertices , they are coloured with either blue or red colour. Without loss of generality, we assume that . Give the alternate colour to that of the vertex . As is coloured with red colour, colour the vertex with blue colour. In general, for , sequentially colour the vertex with the alternate colour to the colour of the vertex . Then observe that contains no induced monochromatic subgraph of order greater than 2 and no monochromatic path in or in can be extended to include any of the vertices of .

Let be the set of all red coloured vertices of and let be the set of all blue coloured vertices of . Then and . Hence is a required -partition of .
Thus, is -partitionable. This completes the induction. Hence every 2-connected graph is -partitionable. ∎

The following Corollary 2.1 is an immediate consequence of Theorem 1.1 and Theorem 2.2.

Corollary 2.1.

Every graph is -partitionable.

It is clear that Corollary 2.1 settles the Path Partition Conjecture affirmatively. Thus, “the Path Partition Conjecture is true”.

The following Theorem 2.3 called “Path Partition Theorem” is a simple implication of Corollary 2.1.

Theorem 2.3 (Path Partition Theorem).

For every graph and for every -tuple of positive integers with and , there exists a partition of such that , for every , .

Proof.

Let be a graph. Consider any -tuple of positive integers with , and . Then by Corollary 2.1, for the pair of positive integers with , where and , there exists a partition of such that and . Consider the graph . Then for the pair of positive integers with , where and , by Corollary 2.1, there exists a partition of such that and . As and , we have and . Similarly, if we consider the pair of positive integers with , where , and , by Corollary 2.1, we get a partition such that and . Continuing this process, finally we get a partition of such that , for every , , where , , and so on. This completes the proof. ∎

Corollary 2.2.

The detour chromatic number for every graph and for every .

Proof.

Let be any graph. For every , consider the -tuple if is a multiple of , while if is not a multiple of , then consider the -tuple , where (mod ). Then, by Path Partition Theorem, there exist a partition , where

such that , for every , . For each , , assign the (distinct) colour to all the vertices in each . Then every monochromatic path in has the order at most . Thus, . ∎

Corollary 2.2 essentially ascertains that “Frick-Bullock Conjecture is true”.
Remark 1: It is clear from the definition of , when , . Thus, by Corollary 2.2, for a graph , . This upper bound for the chromatic number of a graph that is the well known Gallai’s Theorem [11].

3 An Upper Bound for Star Chromatic Number

In this section we obtain an upper bound for star chromatic number as a consequence of path partition theorem.

Theorem 3.1.

Let be a graph. Then the star chromatic number of , .

Proof.

First we prove the result for connected graphs, then the result follows naturally for the disconnected graphs. Let be a connected graph.

Claim 1: There exists a proper -vertex colouring for .

Consider . If is even, say , for some , then consider the -tuple with . By Path Partition Theorem, there exists a partition such that , for every , . Therefore, every induced subgraph , for , is the union of a set of independent vertices and/or a set of independent edges. Thus, it is clear that, for , , each is proper 2-vertex colourable. Properly colour the vertices of each with a distinct pair of colours and , for , . Consequently, this proper 2-vertex colouring of , for all , induces a proper -vertex colouring for the graph . If is odd, say , for some , then consider the -tuple with . Then by Path Partition Theorem there exists a partition such that , for every , and . Consequently, the vertices of each can be properly coloured with a distinct pair of colours and , for , and the vertices of are colored properly with a distinct color . Thus, this proper 2-vertex colouring of , for all , and the proper 1 colouring of induce a proper -vertex colouring for the graph . Hence the Claim 1.

Claim 2:

To prove Claim 2, we show that the vertices of every path of order four is either coloured with 3 or 4 different colours by the above proper -vertex colouring of or if there exists a bicoloured path of order four in by the above proper -vertex colouring of , then those vertices of such a bicoloured path of order four are properly recoloured so that those vertices are coloured with at least three different colours after the recolouring.

Observation 3.1.

As

any path of order four in must contain vertices from at least two of induced subgraphs ’s, where , and when is even, while when is odd, [Hereafter is either or depending on is even or odd respectively]. If any path of order four of contains vertices from three or four of the induced subgraphs ’s then such a path has vertices coloured with three or four colours by the proper -vertex colouring of . Thus, we consider only those paths of order four in having vertices from exactly two of the induced subgraphs ’s, where , for recolouring if it is bicoloured.

Consider any path of order four in having at least one vertex (at most three vertices) in for each , and at least one vertex (at most three vertices) in , for every , .

Case 1. Suppose a path of order four in has one vertex in

and three vertices in , for ,

Then without loss of generality we assume that is one of the vertices of which is in and we assume and are the other three vertices of which are in . Under this situation, in order that the path is to be a path of order four with the vertices , two of the vertices from the three vertices and in must be adjacent in . Since is an independent set of vertices, . As the vertices of each induced subgraph are properly coloured with 2 colours , for , by the proper -vertex colouring, those two adjacent vertices from the three vertices and in should have been coloured with two different colours by the -vertex colouring. In each vertex is coloured with either or by the proper vertex colouring, the vertex is coloured with either or in by the proper -vertex colouring. This implies that the path of order four having the vertices and are coloured with at least three different colours by the proper -vertex colouring of .

Case 2 Suppose a path of order four in has exactly two vertices in

and has exactly two vertices in .

Let and be the two vertices of in and let and be the two vertices of in .

Case 2.1. Suppose either are coloured with two different colours

in or are coloured with two different colours

in by the proper -vertex colouring.

Then the vertices of the path of order four having the vertices and are coloured with three or four different colours by the proper -vertex colouring of .

Case 2.2. Suppose neither the vertices received different colours in

nor the vertices received different colours in

by the proper -vertex colouring.

Then without loss of generality, we assume that received the same colour in and without loss of generality, we assume that received the same colour in by the -vertex colouring. As the vertices of are properly coloured, the vertices and should be non-adjacent in as well as the vertices and should also be non-adjacent in . Since for every , , , is the union of independent vertices and / or independent edges, every vertex in each is of degree either 0 or 1. Suppose either or is of degree 0 in . Then without loss of generality, we assume that is of degree 0 in . Since is not adjacent to any vertex in , recolour the vertex with the colour [Since vertices of are properly coloured with either or colours, this recolouring is possible]. Thus, after this recolouring, the vertices , and of the path have received three different colours. Hence, we assume neither nor is of degree 0 in . Therefore, the degree of each of the vertices and must be of degree 1 in . As vertices of each are properly coloured for , and as the vertices and