Table of Contents
ISRN Discrete Mathematics
Volumeย 2011, Article IDย 430396, 15 pages
http://dx.doi.org/10.5402/2011/430396
Research Article

Chromatic Classes of 2-Connected (๐‘›,๐‘›+4)-Graphs with Exactly Three Triangles and at Least Two Induced 4-Cycles

1Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Johor Campus, Segamat, Malaysia
2Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Malaysia

Received 7 July 2011; Accepted 15 August 2011

Academic Editor: G.ย Isaak

Copyright ยฉ 2011 G. C. Lau and Y. H. Peng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For a graph ๐บ, let ๐‘ƒ(๐บ,๐œ†) be its chromatic polynomial. Two graphs ๐บ and ๐ป are chromatically equivalent, denoted ๐บโˆผ๐ป, if ๐‘ƒ(๐บ,๐œ†)=๐‘ƒ(๐ป,๐œ†). A graph ๐บ is chromatically unique if ๐‘ƒ(๐ป,๐œ†)=๐‘ƒ(๐บ,๐œ†) implies that ๐ปโ‰…๐บ. In this paper, we determine all chromatic equivalence classes of 2-connected (๐‘›,๐‘›+4)-graphs with exactly three triangles and at least two induced 4-cycles. As a byproduct of these, we obtain various new families of ๐œ’-equivalent graphs and ๐œ’-unique graphs.

1. Introduction

Let ๐‘ƒ(๐บ), or simply ๐‘ƒ(๐บ), denote the chromatic polynomial of a simple graph ๐บ. Two graphs ๐บ and ๐ป are chromatically equivalent (simply ๐œ’-equivalent), denoted ๐บโˆผ๐ป, if ๐‘ƒ(๐บ)=๐‘ƒ(๐ป). A graph ๐บ is chromatically unique (simply ๐œ’-unique) if ๐‘ƒ(๐ป)=๐‘ƒ(๐บ) implies that ๐ปโ‰…๐บ. Let โŸจ๐บโŸฉ denote the equivalence class determined by the graph ๐บ under ~. Clearly, ๐บ is ๐œ’-unique if and only if โŸจ๐บโŸฉ={๐บ}. A graph ๐ป is called a relative of ๐บ if there is a sequence of graphs ๐บ=๐ป1,๐ป2,โ€ฆ,๐ป๐‘˜=๐ป such that each ๐ป๐‘– is a ๐พ๐‘Ÿ๐‘–-gluing of some graphs (say ๐‘‹๐‘– and ๐‘Œ๐‘–) and that ๐ป๐‘–+1 is obtained from ๐ป๐‘– by forming another ๐พ๐‘Ÿ๐‘–-gluing of ๐‘‹๐‘– and ๐‘Œ๐‘– for 1โ‰ค๐‘–โ‰ค๐‘˜โˆ’1. We say ๐ป is a graph of type ๐บ if ๐ป is a relative of ๐บ or ๐ปโ‰…๐บ. A family ๐’ฎ of graphs is said to be relative-closed (simply ๐œ’๐‘Ÿ-closed) if(i)no two graphs in ๐’ฎ are relatives of each other, (ii)for any graph ๐บโˆˆ๐’ฎ, ๐‘ƒ(๐ป,๐œ†)=๐‘ƒ(๐บ,๐œ†) implies that ๐ปโˆˆ๐’ฎ or ๐ป is a relative of a graph in ๐‘†.

If ๐’ฎ is a ๐œ’๐‘Ÿ-closed family, then the chromatic equivalence class of each graph in ๐’ฎ can be determined by studying the chromaticity of each graph in ๐’ฎ.

If ๐บ is a graph of order ๐‘› and size ๐‘š, we say ๐บ is an (๐‘›,๐‘š)-graph. The chromatic equivalence classes of 2-connected (๐‘›,๐‘›+๐‘–)-graph have been fully determined for ๐‘–=0,1 in [1, 2] and partially determined for ๐‘–=2,3 in [3โ€“5]. Peng and Lau have also characterized and classified certain chromatic equivalence classes of 2-connected (๐‘›,๐‘›+4)-graph in [6, 7]. In [8], by using the idea of cyclomatic number, the authors obtained the ๐œ’๐‘Ÿ-closed family of 2-connected (๐‘›,๐‘›+4)-graphs with exactly three triangles.

In this paper, all the chromatic equivalence classes of 2-connected (๐‘›,๐‘›+4)-graphs with exactly three triangles and at least two induced ๐ถ4s are determined. As a byproduct of these, we obtain various new families of ๐œ’-equivalent graphs and ๐œ’-unique graphs. The readers may refer to [9] for terms and notation used but not defined here.

2. Notation and Basic Results

Let ๐ถ๐‘› (or ๐‘›-cycle) be the cycle of order ๐‘›. An induced 4-cycle is the cycle ๐ถ4 without chord. The following are some useful known results and techniques for determining the chromatic polynomial of a graph. Throughout this paper, all graphs are assumed to be connected unless otherwise stated.

Lemma 2.1 (Fundamental Reduction Theorem (Whitney [10])). Let ๐บ be a graph and ๐‘’ an edge of ๐บ. Then ๐‘ƒ(๐บ)=๐‘ƒ(๐บโˆ’๐‘’)โˆ’๐‘ƒ(๐บโ‹…๐‘’),(2.1) where ๐บโˆ’๐‘’ is the graph obtained from ๐บ by deleting ๐‘’, and ๐บโ‹…๐‘’ is the graph obtained from ๐บ by identifying the end vertices of ๐‘’.

Let ๐บ1 and ๐บ2 be graphs, each containing a complete subgraph ๐พ๐‘ with ๐‘ vertices. If ๐บ is a graph obtained from ๐บ1 and ๐บ2 by identifying the two subgraphs ๐พ๐‘, then ๐บ is called a ๐พ๐‘-gluing of ๐บ1 and ๐บ2. Note that a ๐พ1-gluing and a ๐พ2-gluing are also called a vertex-gluing and an edge-gluing, respectively.

Lemma 2.2 (Zykov [11]). Let ๐บ be a ๐พ๐‘Ÿ-gluing of ๐บ1 and ๐บ2. Then ๐‘ƒ๎€ท๐บ๐‘ƒ(๐บ)=1๎€ธ๐‘ƒ๎€ท๐บ2๎€ธ๐‘ƒ๎€ท๐พ๐‘Ÿ๎€ธ.(2.2) Lemma 2.2 implies that all ๐พ๐‘Ÿ-gluings of ๐บ1 and ๐บ2 are ๐œ’-equivalent. It follows from Lemma 2.2 that if ๐ป is a relative of ๐บ, then ๐ปโˆผ๐บ.

The following conditions for two graphs ๐บ and ๐ป to be ๐œ’-equivalence are well known (see, e.g., [4]).

Lemma 2.3. Let ๐บ and ๐ป be two ๐œ’-equivalent graphs. Then ๐บ and ๐ป have, respectively, the same number of vertices, edges, and triangles. If both ๐บ and ๐ป do not contain ๐พ4, then they have the same number of induced ๐ถ4s.

A generalized ๐œƒ-graph is a 2-connected graph consisting of three edge-disjoint paths between two vertices of degree 3. All other vertices have degree two. These paths have lengths ๐‘ฅ, ๐‘ฆ and ๐‘ง, respectively, where ๐‘ฅโ‰ฅ๐‘ฆโ‰ฅ๐‘ง. The graph is of order ๐‘ฅ+๐‘ฆ+๐‘งโˆ’1 and size ๐‘ฅ+๐‘ฆ+๐‘ง (see [2]). We will denote ๐พ2 as ๐ถ2 for convenience.

Lemma 2.4. (i)๐‘ƒ๎€ท๐ถ๐‘›๎€ธ=(๐œ†โˆ’1)๐‘›+(โˆ’1)๐‘›(๐œ†โˆ’1),๐‘›โ‰ฅ2,(2.3)(ii)๐‘ƒ๎€ท๐œƒ๐‘ฅ,๐‘ฆ,๐‘ง๎€ธ=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘ƒ๎€ท๐ถ๐‘ฅ+1๎€ธ๐‘ƒ๎€ท๐ถ๐‘ฆ+1๎€ธ๐‘ƒ๎€ท๐ถ๐‘ง+1๎€ธ๐œ†2(๐œ†โˆ’1)2+๐‘ƒ๎€ท๐ถ๐‘ฅ๎€ธ๐‘ƒ๎€ท๐ถ๐‘ฆ๎€ธ๐‘ƒ๎€ท๐ถ๐‘ง๎€ธ๐œ†2,if๐‘ƒ๎€ท๐ถ๐‘งโ‰ 1,๐‘ฅ+1๎€ธ๐‘ƒ๎€ท๐ถ๐‘ฆ+1๎€ธ๐œ†(๐œ†โˆ’1)if๐‘ง=1.(2.4)

Lemma 2.4(i) can be proved by induction while Lemma 2.4(ii) follows from Lemmas 2.1 and 2.2. For integers ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘›, and ๐œ†, let us write ๐‘„๐‘›(๐œ†)=๐‘›โˆ’2๎“๐‘–=0(โˆ’1)๐‘–(๐œ†โˆ’1)๐‘›โˆ’2โˆ’๐‘–,๐‘€๐‘ฅ,๐‘ฆ,๐‘ง(๐œ†)=๐‘„๐‘ฅ+1(๐œ†)๐‘„๐‘ฆ+1(๐œ†)๐‘„๐‘ง+1(๐œ†)+(๐œ†โˆ’1)2๐‘„๐‘ฅ(๐œ†)๐‘„๐‘ฆ(๐œ†)๐‘„๐‘ง(๐œ†).(2.5) Note that when ๐œ†=1, we have ๐‘„๐‘›(1)=(โˆ’1)๐‘› and ๐‘€๐‘ฅ,๐‘ฆ,๐‘ง(1)=(โˆ’1)๐‘ฅ+๐‘ฆ+๐‘ง+1. Lemma 2.4 can then be written as the following lemma.

Lemma 2.5 (see [4]). (i)๐‘ƒ(๐ถ๐‘›)=๐œ†(๐œ†โˆ’1)๐‘„๐‘›(๐œ†) and (ii)๐‘ƒ(๐œƒ๐‘ฅ,๐‘ฆ,๐‘ง)=๐œ†(๐œ†โˆ’1)๐‘€๐‘ฅ,๐‘ฆ,๐‘ง(๐œ†).

We also need the following lemma.

Lemma 2.6 (Whitehead and Zhao [12]). A graph ๐บ contains a cut-vertex if and only if (๐œ†โˆ’1)2โˆฃ๐‘ƒ(๐บ).

Lemma 2.6 also implies that if ๐ปโˆผ๐บ, then ๐ป is 2-connected if and only if ๐บ is so.

3. Classification of Graphs

Let โ„ฑ be the ๐œ’๐‘Ÿ-closed family of 2-connected (๐‘›,๐‘›+4)-graphs with three triangles and at least two induced ๐ถ4s. In [8], we classified all the 31 types of graph ๐นโˆˆโ„ฑ as shown in Figure 1. Since the approach used to classify all the graphs ๐น is rather long and repetitive, we will not discuss it here. The reader may refer to Theorems 1 and 3 in [8] for a detail derivation of the graphs.

430396.fig.001
Figure 1: 31 types of 2-connected (๐‘›,๐‘›+4)-graphs with exactly three triangles and at least two induced 4-cycles. The light lines of the graphs refer to the paths of indicated length.

We are now ready to determine the chromaticity of all 31 types of ๐œ’๐‘Ÿ-closed family of 2-connected (๐‘›,๐‘›+4)-graphs having exactly 3 triangles and at least two induced ๐ถ4s as shown in Figure 1. We first note that if ๐ปโˆผ๐น๐‘–(1โ‰ค๐‘–โ‰ค31) in Figure 1, then ๐ป must be of type ๐น๐‘—(1โ‰ค๐‘—โ‰ค31) in Figure 1 as well. For convenience, we will say that the graph ๐น๐‘–, or any of its relatives, is of type (๐‘–).

In what follows, we will use ๐น๐‘–(๐›ผ), instead of ๐น๐‘–, to denote a graph of type (๐‘–) that has a path of length ๐›ผ. We now present our main results in the following theorem.

Theorem 3.1. (1)๐ปโˆˆโŸจ๐น1โŸฉ if and only if ๐ป is of type ๐น1.
(2)๐ปโˆˆโŸจ๐น2(๐‘Ž)โŸฉ if and only if ๐ป is of type ๐น2(๐‘Ž).
(3)๐ปโˆˆโŸจ๐น3โŸฉ if and only if ๐ป is of type ๐น3.
(4)๐ปโˆˆโŸจ๐น4โŸฉ if and only if ๐ป is of type ๐น4.
(5)๐ปโˆˆโŸจ๐น5โŸฉ if and only if ๐ป is of type ๐น5.
(6)๐ปโˆˆโŸจ๐น6โŸฉ if and only if ๐ปโ‰…๐น6,๐น25 or ๐ป is of type ๐น22(3).
(7)๐ปโˆˆโŸจ๐น7โŸฉ if and only if ๐ปโ‰…๐น7,๐น21,๐น27 or ๐ป is of type ๐น31.
(8)โŸจ๐น8(๐‘)โŸฉ={๐น8(๐‘),๐น28(๐‘)}.
(9)๐น9 is ๐œ’-unique.
(10)โŸจ๐น10โŸฉ={๐น10,๐น29}.
(11)๐ปโˆˆโŸจ๐น11โŸฉ if and only if ๐ป is of type ๐น11, ๐น13(3), or ๐น24.
(12)๐ปโˆˆโŸจ๐น12โŸฉ if and only if ๐ป is of type ๐น12.
(13)๐ปโˆˆโŸจ๐น13(๐‘)โŸฉ if and only if ๐ป is of type ๐น13(๐‘) for ๐‘โ‰ฅ4, and ๐ปโˆˆโŸจ๐น13(3)โŸฉ if and only if ๐ป is of type ๐น11,๐น13(3), or ๐น24.
(14)๐ปโˆˆโŸจ๐น14โŸฉ if and only if ๐ป is of type ๐น14 or ๐น18(3).
(15)๐น15(๐‘‘) is ๐œ’-unique for ๐‘‘โ‰ฅ3, and โŸจ๐น15(2)โŸฉ={๐น15(2),๐น31}.
(16)๐น16 is ๐œ’-unique.
(17)๐น17 is ๐œ’-unique.
(18)๐ปโˆˆโŸจ๐น18(๐‘’)โŸฉ if and only if ๐ป is of type ๐น18(๐‘’) for ๐‘’โ‰ฅ4, and ๐ปโˆˆโŸจ๐น18(3)โŸฉ if and only if ๐ป is of type ๐น14 or ๐น18(3).
(19)๐ปโˆˆโŸจ๐น19โŸฉ if and only if ๐ป is of type ๐น19.
(20)โŸจ๐น20(๐‘“)โŸฉ={๐น20(๐‘“),๐น26(๐‘“)}.
(21)๐ปโˆˆโŸจ๐น21โŸฉ if and only if ๐ปโ‰…๐น7, ๐น21, ๐น27 or ๐ป is of type ๐น31.
(22)๐ปโˆˆโŸจ๐น22(โ„Ž)โŸฉ if and only if ๐ป is of type ๐น22(โ„Ž) for โ„Žโ‰ฅ4, and ๐ปโˆˆโŸจ๐น22(3)โŸฉ if and only if ๐ปโ‰…๐น6,๐น25 or ๐ป is of type ๐น22(3).
(23)๐ปโˆˆโŸจ๐น23โŸฉ if and only if ๐ป is of type ๐น23.
(24)๐ปโˆˆโŸจ๐น24โŸฉ if and only if ๐ป is of type ๐น11, ๐น13(3), or ๐น24.
(25)๐ปโˆˆโŸจ๐น25โŸฉ if and only if ๐ปโ‰…๐น6,๐น25 or ๐ป is of type ๐น22(3).
(26)โŸจ๐น26(๐‘—)โŸฉ={๐น20(๐‘—),๐น26(๐‘—)}.
(27)๐ปโˆˆโŸจ๐น27โŸฉ if and only if ๐ปโ‰…๐น7, ๐น21, ๐น27 or ๐ป is of type ๐น31.
(28)โŸจ๐น28(๐‘˜)โŸฉ={๐น8(๐‘˜),๐น28(๐‘˜)}.
(29)โŸจ๐น29โŸฉ={๐น10,๐น29}.
(30)โŸจ๐น30โŸฉ={๐น15(2),๐น30}.
(31)๐ปโˆˆโŸจ๐น31โŸฉ if and only if ๐ปโ‰…๐น7, ๐น21, ๐น27 or ๐ป is of type ๐น31.

4. Chromatic Polynomials of the Graphs

Before proving our main result, we present here some useful information about the chromatic polynomial of ๐น๐‘– (1โ‰ค๐‘–โ‰ค31). Let ๐‘Š(๐‘›,๐‘˜) denote the graph of order ๐‘› obtained from a wheel ๐‘Š๐‘› by deleting all but ๐‘˜ consecutive spokes. Also let ๐‘Š๐‘š(5,3) denote the graph obtained from ๐‘Š(5,3) by identifying the end-vertices of a path ๐‘ƒ๐‘š to two non-adjacent degree 3 vertices of ๐‘Š(5,3). Using Software Maple or Lemmas 2.1, 2.2 and 2.5, it is easy to obtain the chromatic polynomial of each graph in โ„ฑ as shown in the following lemma.

Lemma 4.1. (1)๐‘ƒ๎€ท๐น1๎€ธ=๐œ†(๐œ†โˆ’1)๐‘1(๐œ†),(4.1) where ๐‘1(๐œ†)=(๐œ†โˆ’2)(๐œ†2โˆ’3๐œ†+3)(๐œ†3โˆ’6๐œ†2+13๐œ†โˆ’11) and ๐‘1(1)=3.
(2)๐‘ƒ๎€ท๐น2๎€ธ=๎€ท๐ถ(๐‘Ž)(๐œ†โˆ’2)๐‘ƒ๐‘Ž+1๎€ธ๐‘ƒ(๐‘Š(5,3))๐œ†(๐œ†โˆ’1)=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)2๎€ท๐œ†2๎€ธ๐‘„โˆ’4๐œ†+5๐‘Ž+1(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘2(๐œ†),(4.2) where ๐‘2(๐œ†)=(๐œ†โˆ’2)2(๐œ†2โˆ’4๐œ†+5)๐‘„๐‘Ž+1(๐œ†) and ๐‘2(1)=(โˆ’1)2(1โˆ’4+5)(โˆ’1)๐‘Ž+1=2(โˆ’1)๐‘Ž+1.
(3)๐‘ƒ๎€ท๐น3๎€ธ=๐œ†(๐œ†โˆ’1)๐‘3(๐œ†),(4.3) where ๐‘3(๐œ†)=(๐œ†โˆ’2)2(๐œ†2โˆ’4๐œ†+5)(๐œ†2โˆ’3๐œ†+3) and ๐‘3(1)=2.
(4)๐‘ƒ๎€ท๐น4๎€ธ=๐œ†(๐œ†โˆ’1)๐‘4(๐œ†),(4.4) where ๐‘4(๐œ†)=(๐œ†โˆ’2)3(๐œ†2โˆ’3๐œ†+3)2 and ๐‘4(1)=โˆ’1.
(5)๐‘ƒ๎€ท๐น5๎€ธ=๐œ†(๐œ†โˆ’1)๐‘5(๐œ†),(4.5) where ๐‘5(๐œ†)=(๐œ†โˆ’2)3(๐œ†3โˆ’5๐œ†2+10๐œ†โˆ’7) and ๐‘5(1)=1.
(6)๐‘ƒ๎€ท๐น6๎€ธ=๐œ†(๐œ†โˆ’1)๐‘6(๐œ†),(4.6) where ๐‘6(๐œ†)=(๐œ†โˆ’2)(๐œ†2โˆ’4๐œ†+5)(๐œ†3โˆ’5๐œ†2+9๐œ†โˆ’7) and ๐‘6(1)=4.
(7)๐‘ƒ๎€ท๐น7๎€ธ=๐œ†(๐œ†โˆ’1)๐‘7(๐œ†),(4.7) where ๐‘7(๐œ†)=(๐œ†โˆ’2)2(๐œ†3โˆ’6๐œ†2+14๐œ†โˆ’13) and ๐‘7(1)=(โˆ’1)2(1โˆ’6+14โˆ’13)=โˆ’4.
(8)๐‘ƒ๎€ท๐น8๎€ธ(๐‘)=(๐œ†โˆ’2)3๐‘ƒ๎€ท๐ถ๐‘+2๎€ธ=โˆ’(๐œ†โˆ’3)๐‘ƒ(๐‘Š(๐‘+3,3))(๐œ†โˆ’2)3๐‘ƒ๎€ท๐ถ๐‘+2๎€ธโˆ’๎€บ๐‘ƒ๎€ท๐ถ(๐œ†โˆ’2)(๐œ†โˆ’3)๐‘+2๎€ธ๎€ท๐ถโˆ’๐‘ƒ๐‘+1๐œ†๎€ธ๎€ป=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)๎€บ๎€ท2๎€ธ๐‘„โˆ’5๐œ†+7๐‘+2(๐œ†)+(๐œ†โˆ’3)๐‘„๐‘+1๎€ป(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘8(๐œ†),(4.8) where ๐‘8(๐œ†)=(๐œ†โˆ’2)[(๐œ†2โˆ’5๐œ†+7)๐‘„๐‘+2(๐œ†)+(๐œ†โˆ’3)๐‘„๐‘+1(๐œ†)] and ๐‘8(1)=(โˆ’1)[3(โˆ’1)๐‘+2+(โˆ’2)(โˆ’1)๐‘+1]=5(โˆ’1)๐‘+1.
(9)๐‘ƒ๎€ท๐น9๎€ธ=๐œ†(๐œ†โˆ’1)๐‘9(๐œ†),(4.9) where ๐‘9(๐œ†)=(๐œ†โˆ’2)2(๐œ†3โˆ’6๐œ†2+14๐œ†โˆ’14) and ๐‘9(1)=โˆ’5.
(10)๐‘ƒ๎€ท๐น10๎€ธ=๐œ†(๐œ†โˆ’1)๐‘10(๐œ†),(4.10) where ๐‘10(๐œ†)=(๐œ†โˆ’2)(๐œ†4โˆ’8๐œ†3+26๐œ†2โˆ’41๐œ†+27) and ๐‘10(1)=โˆ’5.
(11)๐‘ƒ๎€ท๐น11๎€ธ=๐œ†(๐œ†โˆ’1)๐‘11(๐œ†),(4.11) where ๐‘11(๐œ†)=(๐œ†โˆ’2)2(๐œ†4โˆ’7๐œ†3+20๐œ†2โˆ’28๐œ†+17) and ๐‘11(1)=3.
(12)๐‘ƒ๎€ท๐น12๎€ธ=๐œ†(๐œ†โˆ’1)๐‘12(๐œ†),(4.12) where ๐‘12(๐œ†)=(๐œ†โˆ’2)3(๐œ†2โˆ’4๐œ†+6) and ๐‘12(1)=โˆ’3.
(13)๐‘ƒ๎€ท๐น13๎€ธ๎ƒฌ(๐‘)=(๐œ†โˆ’2)(๐œ†โˆ’2)2๐‘ƒ๎€ท๐ถ๐‘+2๎€ธโˆ’๐‘ƒ๎€ท๐พ4๎€ธ๐‘ƒ๎€ท๐ถ๐‘+1๎€ธ๎ƒญ=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)3๐‘ƒ๎€ท๐ถ๐‘+2๎€ธโˆ’(๐œ†โˆ’2)2๎€ท๐ถ(๐œ†โˆ’3)๐‘ƒ๐‘+1๎€ธ=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)2๎€บ(๐œ†โˆ’2)๐‘„๐‘+2(๐œ†)โˆ’(๐œ†โˆ’3)๐‘„๐‘+1๎€ป(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘13(๐œ†),(4.13) where ๐‘13(๐œ†)=(๐œ†โˆ’2)2[(๐œ†โˆ’2)๐‘„๐‘+2(๐œ†)โˆ’(๐œ†โˆ’3)๐‘„๐‘+1(๐œ†)] and ๐‘13(1)=(โˆ’1)2[(โˆ’1)(โˆ’1)๐‘+2โˆ’(โˆ’2)(โˆ’1)๐‘+1]=3(โˆ’1)๐‘+1.
(14)๐‘ƒ๎€ท๐น14๎€ธ=๐œ†(๐œ†โˆ’1)๐‘14(๐œ†),(4.14) where ๐‘14(๐œ†)=(๐œ†โˆ’2)4(๐œ†2โˆ’3๐œ†+4) and ๐‘14(1)=2.
(15)๐‘ƒ๎€ท๐น15๎€ธ=(๐‘‘)(๐œ†โˆ’2)๐‘ƒ(๐‘Š(๐‘‘+4,3))โˆ’(๐œ†โˆ’3)๐‘ƒ(๐‘Š(๐‘‘+3,3))=(๐œ†โˆ’2)2๎€บ๐‘ƒ๎€ท๐ถ๐‘‘+3๎€ธ๎€ท๐ถโˆ’๐‘ƒ๐‘‘+2๎€บ๐‘ƒ๎€ท๐ถ๎€ธ๎€ปโˆ’(๐œ†โˆ’2)(๐œ†โˆ’3)๐‘‘+2๎€ธ๎€ท๐ถโˆ’๐‘ƒ๐‘‘+1๎€บ๎€ธ๎€ป=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)(๐œ†โˆ’2)๐‘„๐‘‘+3(๐œ†)โˆ’(2๐œ†โˆ’5)๐‘„๐‘‘+2(๐œ†)+(๐œ†โˆ’3)๐‘„๐‘‘+1๎€ป(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘15(๐œ†),(4.15) where ๐‘15(๐œ†)=(๐œ†โˆ’2)[(๐œ†โˆ’2)๐‘„๐‘‘+3(๐œ†)โˆ’(2๐œ†โˆ’5)๐‘„๐‘‘+2(๐œ†)+(๐œ†โˆ’3)๐‘„๐‘‘+1(๐œ†)] and ๐‘15(1)=(โˆ’1)[(โˆ’1)(โˆ’1)๐‘‘+3โˆ’(โˆ’3)(โˆ’1)๐‘‘+2+(โˆ’2)(โˆ’1)๐‘‘+1]=6(โˆ’1)๐‘‘+1.
(16)๐‘ƒ๎€ท๐น16๎€ธ=๐œ†(๐œ†โˆ’1)๐‘16(๐œ†),(4.16) where ๐‘16(๐œ†)=(๐œ†โˆ’2)(๐œ†3โˆ’7๐œ†2+19๐œ†โˆ’19) and ๐‘16(1)=6.
(17)๐‘ƒ๎€ท๐น17๎€ธ=๐œ†(๐œ†โˆ’1)๐‘17(๐œ†),(4.17) where ๐‘17(๐œ†)=(๐œ†โˆ’2)(๐œ†4โˆ’8๐œ†3+26๐œ†2โˆ’41๐œ†+25) and ๐‘17(1)=โˆ’3.
(18)๐‘ƒ๎€ท๐น18๎€ธ=๎€บ๎€ท๐ถ(๐‘’)(๐œ†โˆ’2)(๐œ†โˆ’1)๐‘ƒ(๐‘Š(๐‘’+3,3))โˆ’(๐œ†โˆ’2)(๐œ†โˆ’3)๐‘ƒ๐‘’+1๎€ธ๎€ป=(๐œ†โˆ’1)(๐œ†โˆ’2)2๎€บ๐‘ƒ๎€ท๐ถ๐‘’+2๎€ธ๎€ท๐ถโˆ’๐‘ƒ๐‘’+1๎€ธ๎€ปโˆ’(๐œ†โˆ’2)2๎€ท๐ถ(๐œ†โˆ’3)๐‘ƒ๐‘’+1๎€ธ=(๐œ†โˆ’1)(๐œ†โˆ’2)2๐‘ƒ๎€ท๐ถ๐‘’+2๎€ธโˆ’2(๐œ†โˆ’2)3๐‘ƒ๎€ท๐ถ๐‘’+1๎€ธ=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)2๎€บ(๐œ†โˆ’1)๐‘„๐‘’+2(๐œ†)โˆ’2(๐œ†โˆ’2)๐‘„๐‘’+1๎€ป(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘18(๐œ†),(4.18) where ๐‘18(๐œ†)=(๐œ†โˆ’2)2[(๐œ†โˆ’1)๐‘„๐‘’+2(๐œ†)โˆ’2(๐œ†โˆ’2)๐‘„๐‘’+1(๐œ†)] and ๐‘18(1)=(โˆ’1)2[0โˆ’2(โˆ’1)(โˆ’1)๐‘’+1]=2(โˆ’1)๐‘’+1.
(19)๐‘ƒ๎€ท๐น19๎€ธ=๐œ†(๐œ†โˆ’1)๐‘19(๐œ†),(4.19) where ๐‘19(๐œ†)=(๐œ†โˆ’2)2(๐œ†3โˆ’6๐œ†2+14๐œ†โˆ’11) and ๐‘19(1)=โˆ’2.
(20)๐‘ƒ๎€ท๐น20๎€ธ๎€ท๐‘Š(๐‘“)=๐‘ƒ๐‘“+1๎€ธโˆ’๐‘ƒ๎€ท๐ถ(5,3)(๐‘Š(5,3))๐‘ƒ๐‘“+1๎€ธ๎€ท๐œƒ๐œ†(๐œ†โˆ’1)=(๐œ†โˆ’2)๐‘ƒ๐‘“+1,2,2๎€ธโˆ’(๐œ†โˆ’2)2๐‘ƒ๎€ท๐ถ๐‘“+2๎€ธ๎€ท๐œ†โˆ’(๐œ†โˆ’2)2๎€ธ๐‘ƒ๎€ท๐ถโˆ’4๐œ†+5๐‘“+1๎€ธ๎€บ=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)๐‘€๐‘“+1,2,2(๐œ†)โˆ’(๐œ†โˆ’2)2๐‘„๐‘“+2๎€ท๐œ†(๐œ†)โˆ’(๐œ†โˆ’2)2๎€ธ๐‘„โˆ’4๐œ†+5๐‘“+1๎€ป(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘20(๐œ†),(4.20) where ๐‘20(๐œ†)=(๐œ†โˆ’2)๐‘€๐‘“+1,2,2(๐œ†)โˆ’(๐œ†โˆ’2)2๐‘„๐‘“+2(๐œ†)โˆ’(๐œ†โˆ’2)(๐œ†2โˆ’4๐œ†+5)๐‘„๐‘“+1(๐œ†) and ๐‘20(1)=(โˆ’1)(โˆ’1)๐‘“โˆ’(โˆ’1)๐‘“โˆ’(โˆ’1)(2)(โˆ’1)๐‘“+1=4(โˆ’1)๐‘“+1.
(21)๐‘ƒ๎€ท๐น21๎€ธ=๐œ†(๐œ†โˆ’1)๐‘21(๐œ†),(4.21) where ๐‘21(๐œ†)=(๐œ†โˆ’2)2(๐œ†3โˆ’6๐œ†2+14๐œ†โˆ’13) and ๐‘21(1)=โˆ’4.
(22)๐‘ƒ๎€ท๐น22๎€ธ=(โ„Ž)๐‘ƒ(๐‘Š(โ„Ž+3,3))๐‘ƒ(๐‘Š(5,3))๐‘ƒ๎€ท๐พ3๎€ธ๎€ท๐œ†=(๐œ†โˆ’2)2๐‘ƒ๎€ท๐ถโˆ’4๐œ†+5๎€ธ๎€บโ„Ž+2๎€ธ๎€ท๐ถโˆ’๐‘ƒโ„Ž+1๎€ท๐œ†๎€ธ๎€ป=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)2๐‘„โˆ’4๐œ†+5๎€ธ๎€บโ„Ž+2(๐œ†)โˆ’๐‘„โ„Ž+1๎€ป(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘22(๐œ†),(4.22) where ๐‘22(๐œ†)=(๐œ†โˆ’2)(๐œ†2โˆ’4๐œ†+5)[๐‘ƒ(๐‘„โ„Ž+2(๐œ†)โˆ’๐‘ƒ(๐‘„โ„Ž+1(๐œ†)] and ๐‘22(1)=(โˆ’1)(2)[(โˆ’1)โ„Ž+2โˆ’(โˆ’1)โ„Ž+1]=4(โˆ’1)โ„Ž+1.
(23)๐‘ƒ๎€ท๐น23๎€ธ=๐œ†(๐œ†โˆ’1)๐‘23(๐œ†),(4.23) where ๐‘23(๐œ†)=(๐œ†โˆ’2)(๐œ†2โˆ’4๐œ†+5)2 and ๐‘23(1)=โˆ’4.
(24)๐‘ƒ๎€ท๐น24๎€ธ=๐œ†(๐œ†โˆ’1)๐‘24(๐œ†),(4.24) where ๐‘24(๐œ†)=(๐œ†โˆ’2)2(๐œ†4โˆ’7๐œ†3+20๐œ†2โˆ’28๐œ†+17) and ๐‘24(1)=3.
(25)๐‘ƒ๎€ท๐น25๎€ธ=๐œ†(๐œ†โˆ’1)๐‘25(๐œ†),(4.25) where ๐‘25(๐œ†)=(๐œ†โˆ’2)(๐œ†2โˆ’4๐œ†+5)(๐œ†3โˆ’5๐œ†2+9๐œ†โˆ’7) and ๐‘25(1)=4.
(26)๐‘ƒ๎€ท๐น26๎€ธ๎€ท๐‘Š(๐‘—)=๐‘ƒ๐‘—+1๎€ธโˆ’๐‘ƒ๎€ท๐ถ(5,3)(๐‘Š(5,3))๐‘ƒ๐‘—+1๎€ธ๎€ท๐œƒ๐œ†(๐œ†โˆ’1)=(๐œ†โˆ’2)๐‘ƒ๐‘—+1,2,2๎€ธโˆ’(๐œ†โˆ’2)2๐‘ƒ๎€ท๐ถ๐‘—+2๎€ธ๎€ท๐œ†โˆ’(๐œ†โˆ’2)2๎€ธ๐‘ƒ๎€ท๐ถโˆ’4๐œ†+5๐‘—+1๎€ธ๎€บ=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)๐‘€๐‘—+1,2,2(๐œ†)โˆ’(๐œ†โˆ’2)2๐‘„๐‘—+2๎€ท๐œ†(๐œ†)โˆ’(๐œ†โˆ’2)2๎€ธ๐‘„โˆ’4๐œ†+5๐‘—+1๎€ป(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘26(๐œ†),(4.26) where ๐‘26(๐œ†)=(๐œ†โˆ’2)๐‘€๐‘—+1,2,2(๐œ†)โˆ’(๐œ†โˆ’2)2๐‘„๐‘—+2(๐œ†)โˆ’(๐œ†โˆ’2)(๐œ†2โˆ’4๐œ†+5)๐‘„๐‘—+1(๐œ†) and ๐‘26(1)=(โˆ’1)(โˆ’1)๐‘—โˆ’(โˆ’1)๐‘—โˆ’(โˆ’1)(2)(โˆ’1)๐‘—+1=4(โˆ’1)๐‘—+1.
(27)๐‘ƒ๎€ท๐น27๎€ธ=๐œ†(๐œ†โˆ’1)๐‘27(๐œ†)(4.27) where ๐‘27(๐œ†)=(๐œ†โˆ’2)2(๐œ†3โˆ’6๐œ†2+14๐œ†โˆ’13) and ๐‘27(1)=โˆ’4.
(28)๐‘ƒ๎€ท๐น28๎€ธ(๐‘˜)=(๐œ†โˆ’2)3๐‘ƒ๎€ท๐ถ๐‘˜+2๎€ธ=โˆ’(๐œ†โˆ’3)๐‘ƒ(๐‘Š(๐‘˜+3,3))(๐œ†โˆ’2)3๐‘ƒ๎€ท๐ถ๐‘˜+2๎€ธโˆ’๎€บ๐‘ƒ๎€ท๐ถ(๐œ†โˆ’2)(๐œ†โˆ’3)๐‘˜+2๎€ธ๎€ท๐ถโˆ’๐‘ƒ๐‘˜+1๐œ†๎€ธ๎€ป=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)๎€บ๎€ท2๎€ธ๐‘„โˆ’5๐œ†+7๐‘˜+2(๐œ†)+(๐œ†โˆ’3)๐‘„๐‘˜+1๎€ป(๐œ†)=๐œ†(๐œ†โˆ’1)๐‘28(๐œ†),(4.28) where ๐‘28(๐œ†)=(๐œ†โˆ’2)[(๐œ†2โˆ’5๐œ†+7)๐‘„๐‘˜+2(๐œ†)+(๐œ†โˆ’3)๐‘„๐‘˜+1(๐œ†)] and ๐‘28(1)=(โˆ’1)[3(โˆ’1)๐‘˜+2+(โˆ’2)(โˆ’1)๐‘˜+1]=5(โˆ’1)๐‘˜+1.
(29)๐‘ƒ๎€ท๐น29๎€ธ=๐œ†(๐œ†โˆ’1)๐‘29(๐œ†),(4.29) where ๐‘29(๐œ†)=(๐œ†โˆ’2)(๐œ†4โˆ’8๐œ†3+26๐œ†2โˆ’41๐œ†+27) and ๐‘29(1)=โˆ’5.
(30)๐‘ƒ๎€ท๐น30๎€ธ=๐œ†(๐œ†โˆ’1)๐‘30(๐œ†),(4.30) where ๐‘30(๐œ†)=(๐œ†โˆ’2)(๐œ†4โˆ’8๐œ†3+26๐œ†2โˆ’42๐œ†+29) and ๐‘30(1)=โˆ’6.
(31)๐‘ƒ๎€ท๐น31๎€ธ=๐œ†(๐œ†โˆ’1)๐‘31(๐œ†),(4.31) where ๐‘31(๐œ†)=(๐œ†โˆ’2)2(๐œ†3โˆ’6๐œ†2+14๐œ†โˆ’13) and ๐‘31(1)=โˆ’4.

Lemma 4.2. Let โ„ฑ1={๐น4,๐น5},โ„ฑ2={๐น2,๐น3,๐น14,๐น18,๐น19}, โ„ฑ3={๐น1,๐น11,๐น12,๐น13,๐น17,๐น24}, โ„ฑ4={๐น6,๐น7,๐น20,๐น21,๐น22,๐น23,๐น25, ๐น26,๐น27,๐น31}, โ„ฑ5={๐น8,๐น9,๐น10,๐น28,๐น29}, and โ„ฑ6={๐น15,๐น16,๐น30}. Then, for each ๐นโˆˆโ„ฑ๐‘–, ๐‘–=1,2,3,4,5,6, ๐ปโˆผ๐น implies that ๐ป must be of type ๐น or ๐น๎…ž for an ๐น๎…ž in โ„ฑ๐‘–.

Proof. It follows directly from Lemma 4.1 that if ๐‘–โ‰ ๐‘—, ๐น๐‘โˆˆโ„ฑ๐‘– and ๐น๐‘žโˆˆโ„ฑ๐‘—, then |๐‘๐‘(1)|=๐‘–โ‰ ๐‘—=|๐‘๐‘ž(1)|.

From Lemmas 2.3 and 4.1, we also get the following lemma directly.

Lemma 4.3. (1)๐น6โˆผ๐น25.
(2)๐น7โˆผ๐น21โˆผ๐น27โˆผ๐น31.
(3)๐น8(๐‘)โˆผ๐น28(๐‘˜) if and only if ๐‘=๐‘˜.
(4)๐น10โˆผ๐น29.
(5)๐น11โˆผ๐น24.
(6)๐น20(๐‘“)โˆผ๐น26(๐‘—) if and only if ๐‘“=๐‘—.

5. Proof of the Main Theorem

We are now ready to prove our main theorem.

(1) Let ๐ปโˆผ๐น1. By Lemma 4.2, ๐ป is of type (1), (11), (12), (13), (17), or (24). If ๐ป=๐น1, then ๐ป is of type ๐น1. Lemma 4.1 further implies that ๐‘ƒ(๐น1,๐œ†)โ‰ ๐‘ƒ(๐น๐‘–,๐œ†), ๐‘–=11,12,17,24. Hence, ๐ป cannot be of type (11), (12), (17), or (24). If ๐ป=๐น13(๐‘), by Lemma 2.3, ๐‘=3. Using Software Maple, we have ๐‘ƒ๎€ท๐น13๎€ธ(3)=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)2๎€ท๐œ†4โˆ’7๐œ†3+20๐œ†2๎€ธ๎€ท๐œ†โˆ’28๐œ†+17โ‰ (๐œ†โˆ’2)2๐œ†โˆ’3๐œ†+3๎€ธ๎€ท3โˆ’6๐œ†2๎€ธ๎€ท๐น+13๐œ†โˆ’11=๐‘ƒ1๎€ธ.(5.1) Thus, ๐ป must be of type ๐น1.

(2) Let ๐ปโˆผ๐น2. By Lemma 4.2, ๐ป is of type (2), (3), (14), (18), or (19). If ๐ป=๐น2(๐‘Ž๎…ž), then by Lemma 2.3, ๐‘Ž๎…ž=๐‘Ž. Thus, ๐ป must be of type ๐น2. Since ๐น2(๐‘Ž) has two induced ๐ถ4s while each of ๐น3 and ๐น19 has at least three induced ๐ถ4s, by Lemma 2.3, ๐ป cannot be of type (3) or (19). Since ๐‘ƒ(๐น14) is divisible by (๐œ†โˆ’2)4 but not ๐‘ƒ(๐น2(๐‘Ž)), ๐ป cannot be of type (14). If ๐ป=๐น18(๐‘’), then by Lemma 2.3, ๐‘’=๐‘Ž. Note that ๐‘ƒ๎€ท๐น2๎€ธ(๐‘Ž)=(๐œ†โˆ’1)(๐œ†โˆ’2)3๐‘ƒ๎€ท๐ถ๐‘Ž+1๎€ธโˆ’(๐œ†โˆ’2)2๎€ท๐ถ(๐œ†โˆ’3)๐‘ƒ๐‘Ž+1๎€ธ,๐‘ƒ๎€ท๐น18๎€ธ=(๐‘Ž)(๐œ†โˆ’1)(๐œ†โˆ’2)๐‘ƒ(๐‘Š(๐‘Ž+3,3))โˆ’(๐œ†โˆ’2)2๎€ท๐ถ(๐œ†โˆ’3)๐‘ƒ๐‘Ž+1๎€ธ.(5.2) This implies that (๐œ†โˆ’2)2๐‘ƒ(๐ถ๐‘Ž+1)=๐‘ƒ(๐‘Š(๐‘Ž+3,3)), a contradiction since ๐‘ƒ(๐‘Š(๐‘Ž+3,3)) is not divisible by (๐œ†โˆ’2)2. Thus, ๐ปโˆˆโŸจ๐น2(๐‘Ž)โŸฉ if and only if ๐ป is of type ๐น2(๐‘Ž).

(3) Let ๐ปโˆผ๐น3. By Lemma 4.2 and the above result, ๐ป is of type (3), (14), (18), or (19). If ๐ป=๐น3, then ๐ป is of type ๐น3. By Lemma 4.1, ๐น3ฬธโˆผ๐น14 and ๐น19. If ๐ป=๐น18(๐‘’), by Lemma 2.3, ๐‘’=3. Using Software Maple, we have ๐‘ƒ๎€ท๐น18๎€ธ(3),๐œ†=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)4๎€ท๐œ†2๎€ธโ‰ โˆ’3๐œ†+4(๐œ†โˆ’2)2๎€ท๐œ†2๐œ†โˆ’4๐œ†+5๎€ธ๎€ท2๎€ธ๎€ท๐นโˆ’3๐œ†+3=๐‘ƒ3๎€ธ.,๐œ†(5.3) Thus, ๐ป must be of type ๐น3.

(4) Let ๐ปโˆผ๐น4. By Lemma 4.2, ๐ป is of type (4) or (5). If follows directly from Lemma 4.1 that ๐น4ฬธโˆผ๐น5. Thus, ๐ป must be of type ๐น4.

(5) Let ๐ปโˆผ๐น5. By Lemma 4.2 and the above result, ๐ป must be of type (5). Thus, ๐ป must be of type ๐น5.

(6) By Lemma 4.2, ๐ป is of type (6), (7), (20), (21), (22), (23), (25), (26), (27), or (31). If ๐ป=๐น6, then ๐ปโ‰…๐น6. Note that Lemma 4.1 implies that ๐น6ฬธโˆผ๐น๐‘–, ๐‘–=7,21,23,27,31. If ๐ป=๐น20(๐‘“), ๐น22(โ„Ž), or ๐น26(๐‘—), by Lemma 2.3, ๐‘“=โ„Ž=๐‘—=3. Using Software Maple, we have ๐‘ƒ๎€ท๐น20๎€ธ๎€ท๐น(3),๐œ†=๐‘ƒ26๎€ธ(3),๐œ†=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)2๎€ท๐œ†4โˆ’7๐œ†3+20๐œ†2๎€ธ๎€ท๐œ†โˆ’28๐œ†+18โ‰ ๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)2๐œ†โˆ’4๐œ†+5๎€ธ๎€ท3โˆ’5๐œ†2๎€ธ๎€ท๐น+9๐œ†โˆ’7=๐‘ƒ22๎€ธ๎€ท๐น(3),๐œ†=๐‘ƒ6๎€ธ.,๐œ†(5.4) Thus, by Lemma 4.3, ๐ปโˆˆโŸจ๐น6โŸฉ if and only if ๐ปโ‰…๐น6,๐น25 or of type ๐น22(3).

(7) Let ๐ปโˆผ๐น7. By Lemma 4.2 and the above results, ๐ป is of type (7), (20), (21), (22) where โ„Žโ‰ฅ4, (23), (26), (27), or (31). If ๐ป=๐น๐‘–, ๐‘–=7,21,27,31, Lemma 4.3 implies that ๐ปโ‰…๐น7, ๐น21, ๐น27, or ๐ป is of type ๐น31. Lemma 4.1 further implies that ๐ป cannot be of type (20), (22), (23), or (26). Thus, ๐ปโˆˆโŸจ๐น7โŸฉ if and only if ๐ปโ‰…๐น7, ๐น21, ๐น27, or ๐ป is of type ๐น31.

(8) Let ๐ปโˆผ๐น8(๐‘). By Lemma 4.2, ๐ป is of type (8), (9), (10), (28), or (29). If ๐ป=๐น8(๐‘๎…ž), by Lemma 2.3, ๐‘๎…ž=๐‘. Thus, ๐ปโ‰…๐น8(๐‘). Since ๐น8(๐‘) is of order at least 8 but ๐น๐‘–, ๐‘–=9,10,29 is of order 7, by Lemma 2.3, ๐‘ƒ(๐น8(๐‘))โ‰ ๐‘ƒ(๐น๐‘–), ๐‘–=9,10,29. By Lemma 4.3, ๐‘ƒ(๐น8(๐‘))=๐‘ƒ(๐น28(๐‘)). Hence, โŸจ๐น8(๐‘)โŸฉ={๐น8(๐‘),๐น28(๐‘)}.

(9) Let ๐ปโˆผ๐น9. By Lemma 4.2 and the above results, ๐ป is of type (9), (10), or (29). By Lemma 4.1, ๐น9ฬธโˆผ๐น10,๐น29. Thus, ๐ปโ‰…๐น9 and ๐น9 is ๐œ’-unique.

(10) Let ๐ปโˆผ๐น10. By Lemma 4.2 and the above result, ๐ป is of type (10) or (29). By Lemma 4.3, โŸจ๐น10โŸฉ={๐น10,๐น29}.

(11) Let ๐ปโˆผ๐น11. By Lemma 4.2 and the above result, ๐ป is of type (11), (12), (13), (17), or (24). If ๐ป=๐น11 or ๐น24, by Lemma 4.3, ๐ป must be of type ๐น11 or ๐น24. Lemma 4.1 further implies that ๐‘ƒ(๐น11,๐œ†)โ‰ ๐‘ƒ(๐น12,๐œ†) and ๐‘ƒ(๐น17,๐œ†). Hence, ๐ป cannot be of type (12) or (17). If ๐ป=๐น13(๐‘), Lemma 2.3 implies that ๐‘=3. Using Software Maple, we have ๐‘ƒ๎€ท๐น13๎€ธ(3),๐œ†=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)2๎€ท๐œ†4โˆ’7๐œ†3+20๐œ†2๎€ธ๎€ท๐นโˆ’28๐œ†+17=๐‘ƒ11๎€ธ.,๐œ†(5.5) Hence, ๐ปโˆˆโŸจ๐น11โŸฉ if and only if ๐ป is of type ๐น11, ๐น13(3), or ๐น24.

(12) Let ๐ปโˆผ๐น12. By Lemma 4.2 and the above result, ๐ป is of type (12), (13) with ๐‘โ‰ฅ4 or (17). Since ๐น12 and ๐น13(๐‘) have different order, Lemma 2.3 implies that ๐น12ฬธโˆผ๐น13. Lemma 4.1 also implies that ๐น12ฬธโˆผ๐น17. Thus, ๐ป must be of type ๐น12.

(13) Let ๐ปโˆผ๐น13(๐‘),๐‘โ‰ฅ4. By Lemma 4.2 and the above result, ๐ป is of type (13) with ๐‘โ‰ฅ4 or (17). If ๐ป=๐น13(๐‘๎…ž), then ๐‘๎…ž=๐‘. Since ๐น13(๐‘) and ๐น17 have different order, Lemma 2.3 implies that ๐น13ฬธ(๐‘)โˆผ๐น17. Thus, ๐ปโˆˆโŸจ๐น13(๐‘)โŸฉ if and only if ๐ป is of type ๐น13(๐‘) for ๐‘โ‰ฅ4 and ๐ปโˆˆโŸจ๐น13(3)โŸฉ if and only if ๐ป is of type ๐น11, ๐น13(3), or ๐น24.

(14) Let ๐ปโˆผ๐น14. By Lemma 4.2 and the above result, ๐ป is of type (14), (18) or (19). If ๐ป=๐น14, then ๐ป is of type ๐น14. If ๐ป=๐น18(๐‘’), by Lemma 2.3, ๐‘’=3. Using Software Maple, we have ๐‘ƒ๎€ท๐น18๎€ธ(3),๐œ†=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)4๎€ท๐œ†2๎€ธ๎€ท๐นโˆ’3๐œ†+4=๐‘ƒ14๎€ธ.,๐œ†(5.6) By Lemma 4.1, we also have ๐น14ฬธโˆผ๐น19. Hence, ๐ปโˆˆโŸจ๐น14โŸฉ if and only if ๐ป is of type ๐น14 or ๐น18(3).

(15) Let ๐ปโˆผ๐น15(๐‘‘). By Lemma 4.2, ๐ป must be of type (15), (16), or (30). If ๐ป=๐น15(๐‘‘๎…ž), by Lemma 2.3, ๐‘‘๎…ž=๐‘‘. Thus, ๐ปโ‰…๐น15. Since ๐น16 has exactly six induced ๐ถ4s while ๐น15(๐‘‘) has only two induced ๐ถ4s, by Lemma 2.3, ๐ป cannot be of type (16). If ๐ป=๐น31, by Lemma 2.3, ๐‘‘=2. Using Software Maple, we have ๐‘ƒ๎€ท๐น15๎€ธ๎€ท๐œ†(2)=๐œ†(๐œ†โˆ’1)(๐œ†โˆ’2)4โˆ’8๐œ†3+26๐œ†2๎€ธ๎€ท๐นโˆ’42๐œ†+29=๐‘ƒ30๎€ธ.(5.7) Thus, โŸจ๐น15(2)โŸฉ={๐น15(2),๐น30} and ๐น15(๐‘‘) is ๐œ’-unique for ๐‘‘โ‰ฅ3.

(16) Let ๐ปโˆผ๐น16. By Lemma 4.2 and the above results, ๐ปโ‰…๐น16. Thus, ๐น16 is ๐œ’-unique.

(17) Let ๐ปโˆผ๐น17. By Lemma 4.2 and the above results, ๐ปโ‰…๐น17. Thus, ๐น17 is ๐œ’-unique.

(18) Let ๐ปโˆผ๐น18(๐‘’), ๐‘’โ‰ฅ4. By Lemma 4.2 and the above results, ๐ป must be of of type (18) with ๐‘’โ‰ฅ4, or (19). If ๐ป=๐น18(๐‘’๎…ž), Lemma 2.3 implies that ๐‘’๎…ž=๐‘’. Since ๐น18(๐‘’) and ๐น19 are of different order, it follows that ๐ป cannot be of type (19). Thus, ๐ปโˆˆโŸจ๐น18(๐‘’)โŸฉ if and only if ๐ป is of type ๐น18(๐‘’) for ๐‘’โ‰ฅ4, and ๐ปโˆˆโŸจ๐น18(3)โŸฉ if and only if ๐ป is of type ๐น14 or ๐น18(3).

(19) Let ๐ปโˆผ๐น19. By Lemma 4.2 and the above results, ๐ป must be of type ๐น19.

(20) Let ๐ปโˆผ๐น20(๐‘“). By Lemma 4.2 and the above results, ๐ป must be of type (20), (22) where โ„Žโ‰ฅ4, (23) or (26). If ๐ป=๐น20(๐‘“๎…ž), Lemma 2.3 implies that ๐‘“๎…ž=๐‘“. If ๐ป=๐น22(โ„Ž), Lemma 2.3 implies that โ„Ž=๐‘“. Note that ๐‘ƒ๎€ท๐น20๎€ธ=๐‘ƒ๎€ท๐น(๐‘“)(๐œ†โˆ’1)๐‘ƒ(๐‘Š(๐‘“+4,4))โˆ’(๐œ†โˆ’3)๐‘ƒ(๐‘Š(๐‘“+3,3)),22(๎€ธ๐‘“)=(๐œ†โˆ’1)(๐œ†โˆ’2)๐‘ƒ(๐‘Š(๐‘“+3,3))โˆ’(๐œ†โˆ’3)๐‘ƒ(๐‘Š(๐‘“+3,3)).(5.8) This implies that ๐‘ƒ(๐‘Š(๐‘“+4,4))=(๐œ†โˆ’2)๐‘ƒ(๐‘Š(๐‘“+3,3)), a contradiction since ๐‘ƒ(๐‘Š(๐‘“+4,4)) is not divisible by (๐œ†โˆ’2)2 but (๐œ†โˆ’2)๐‘ƒ(๐‘Š(๐‘“+3,3)) is divisible by (๐œ†โˆ’2)2. Since ๐น20 and ๐น23 are of different order, Lemma 2.3 further implies that ๐ป cannot be of type (23). Lemma 4.3 then implies that โŸจ๐น20(๐‘“)โŸฉ={๐น20(๐‘“),๐น26(๐‘“)}.

(21) The result follows directly from (7) above.

(22) Let ๐ปโˆผ๐น22(โ„Ž), โ„Žโ‰ฅ4. By Lemma 4.2 and the above result, ๐ป is of type (22) with โ„Žโ‰ฅ4, or (23). If ๐ป=๐น22(โ„Ž๎…ž), Lemma 2.3 implies that โ„Ž๎…ž=โ„Ž. Since ๐น22(โ„Ž) and ๐น23 are of different order, Lemma 2.3 further implies that ๐ป cannot be of type (23). Thus, ๐ปโˆˆโŸจ๐น22(โ„Ž)โŸฉ if and only if ๐ป is of type ๐น22(โ„Ž) for โ„Žโ‰ฅ4, and ๐ปโˆˆโŸจ๐น22(3)โŸฉ if and only if ๐ปโ‰…๐น6, ๐น25 or ๐ป is of type ๐น22(3).

(23) Let ๐ปโˆผ๐น23. By Lemma 4.2 and the above results, ๐ป must be of type ๐น23. Thus, ๐ปโˆˆโŸจ๐น23โŸฉ if and only if ๐ป is of type ๐น23.

(24) The result follows directly from (11) above.

(25) The result follows directly from (6) above.

(26) The result follows directly from (20) above.

(27) The result follows directly from (7) above.

(28) The result follows directly from (8) above.

(29) The result follows directly from (10) above.

(30) The result follows directly from (15) above.

(31) The result follows directly from (7) above.

This completes the proof of our main theorem.

6. Further Research

The above results and the main results in [6, 7] completely determined the chromaticity of all 2-connected (๐‘›,๐‘›+4)-graphs with (i) exactly 3 triangles (and at least one induced 4-cycle) and (ii) at least 4 triangles. However, the study of the chromaticity of 2-connected (๐‘›,๐‘›+4)-graphs with exactly 3 triangles is far from completion although all 23 ๐œ’๐‘Ÿ-closed families of such graphs have been obtained in [8] as shown in Figure 2. Base on the above results, it is expected that many different families of 2-connected (๐‘›,๐‘›+4)-graphs with exactly 3 triangles are ๐œ’-equivalent. Perhaps, the approach used in the study of the chromaticity of ๐พ4-homeomorphs (see [13]) or a more efficient approach of comparing the chromatic polynomials of graphs can be applied in solving the following problem.

430396.fig.002
Figure 2: Relative-closed family of 2-connected (๐‘›,๐‘›+4)-graphs with exactly 3 triangles. The light lines of the graphs refer to the paths with edges not belong to any triangles.

Problem 1. Determine the chromatic uniqueness of all 2-connected (๐‘›,๐‘›+4)-graphs with exactly 3 triangles.

Acknowledgment

The authors would like to thank the referee for the valuable comments.

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