Abstract

For a graph 𝐺, let 𝑃(𝐺,𝜆) be its chromatic polynomial. Two graphs 𝐺 and 𝐻 are chromatically equivalent, denoted 𝐺𝐻, if 𝑃(𝐺,𝜆)=𝑃(𝐻,𝜆). A graph 𝐺 is chromatically unique if 𝑃(𝐻,𝜆)=𝑃(𝐺,𝜆) implies that 𝐻𝐺. In this paper, we determine all chromatic equivalence classes of 2-connected (𝑛,𝑛+4)-graphs with exactly three triangles and at least two induced 4-cycles. As a byproduct of these, we obtain various new families of 𝜒-equivalent graphs and 𝜒-unique graphs.

1. Introduction

Let 𝑃(𝐺), or simply 𝑃(𝐺), denote the chromatic polynomial of a simple graph 𝐺. Two graphs 𝐺 and 𝐻 are chromatically equivalent (simply 𝜒-equivalent), denoted 𝐺𝐻, if 𝑃(𝐺)=𝑃(𝐻). A graph 𝐺 is chromatically unique (simply 𝜒-unique) if 𝑃(𝐻)=𝑃(𝐺) implies that 𝐻𝐺. Let 𝐺 denote the equivalence class determined by the graph 𝐺 under ~. Clearly, 𝐺 is 𝜒-unique if and only if 𝐺={𝐺}. A graph 𝐻 is called a relative of 𝐺 if there is a sequence of graphs 𝐺=𝐻1,𝐻2,,𝐻𝑘=𝐻 such that each 𝐻𝑖 is a 𝐾𝑟𝑖-gluing of some graphs (say 𝑋𝑖 and 𝑌𝑖) and that 𝐻𝑖+1 is obtained from 𝐻𝑖 by forming another 𝐾𝑟𝑖-gluing of 𝑋𝑖 and 𝑌𝑖 for 1𝑖𝑘1. We say 𝐻 is a graph of type 𝐺 if 𝐻 is a relative of 𝐺 or 𝐻𝐺. A family 𝒮 of graphs is said to be relative-closed (simply 𝜒𝑟-closed) if(i)no two graphs in 𝒮 are relatives of each other, (ii)for any graph 𝐺𝒮, 𝑃(𝐻,𝜆)=𝑃(𝐺,𝜆) implies that 𝐻𝒮 or 𝐻 is a relative of a graph in 𝑆.

If 𝒮 is a 𝜒𝑟-closed family, then the chromatic equivalence class of each graph in 𝒮 can be determined by studying the chromaticity of each graph in 𝒮.

If 𝐺 is a graph of order 𝑛 and size 𝑚, we say 𝐺 is an (𝑛,𝑚)-graph. The chromatic equivalence classes of 2-connected (𝑛,𝑛+𝑖)-graph have been fully determined for 𝑖=0,1 in [1, 2] and partially determined for 𝑖=2,3 in [35]. Peng and Lau have also characterized and classified certain chromatic equivalence classes of 2-connected (𝑛,𝑛+4)-graph in [6, 7]. In [8], by using the idea of cyclomatic number, the authors obtained the 𝜒𝑟-closed family of 2-connected (𝑛,𝑛+4)-graphs with exactly three triangles.

In this paper, all the chromatic equivalence classes of 2-connected (𝑛,𝑛+4)-graphs with exactly three triangles and at least two induced 𝐶4s are determined. As a byproduct of these, we obtain various new families of 𝜒-equivalent graphs and 𝜒-unique graphs. The readers may refer to [9] for terms and notation used but not defined here.

2. Notation and Basic Results

Let 𝐶𝑛 (or 𝑛-cycle) be the cycle of order 𝑛. An induced 4-cycle is the cycle 𝐶4 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 𝐺 be a graph and 𝑒 an edge of 𝐺. Then 𝑃(𝐺)=𝑃(𝐺𝑒)𝑃(𝐺𝑒),(2.1) where 𝐺𝑒 is the graph obtained from 𝐺 by deleting 𝑒, and 𝐺𝑒 is the graph obtained from 𝐺 by identifying the end vertices of 𝑒.

Let 𝐺1 and 𝐺2 be graphs, each containing a complete subgraph 𝐾𝑝 with 𝑝 vertices. If 𝐺 is a graph obtained from 𝐺1 and 𝐺2 by identifying the two subgraphs 𝐾𝑝, then 𝐺 is called a 𝐾𝑝-gluing of 𝐺1 and 𝐺2. Note that a 𝐾1-gluing and a 𝐾2-gluing are also called a vertex-gluing and an edge-gluing, respectively.

Lemma 2.2 (Zykov [11]). Let 𝐺 be a 𝐾𝑟-gluing of 𝐺1 and 𝐺2. Then 𝑃𝐺𝑃(𝐺)=1𝑃𝐺2𝑃𝐾𝑟.(2.2) Lemma 2.2 implies that all 𝐾𝑟-gluings of 𝐺1 and 𝐺2 are 𝜒-equivalent. It follows from Lemma 2.2 that if 𝐻 is a relative of 𝐺, then 𝐻𝐺.

The following conditions for two graphs 𝐺 and 𝐻 to be 𝜒-equivalence are well known (see, e.g., [4]).

Lemma 2.3. Let 𝐺 and 𝐻 be two 𝜒-equivalent graphs. Then 𝐺 and 𝐻 have, respectively, the same number of vertices, edges, and triangles. If both 𝐺 and 𝐻 do not contain 𝐾4, then they have the same number of induced 𝐶4s.

A generalized 𝜃-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 𝑥, 𝑦 and 𝑧, respectively, where 𝑥𝑦𝑧. The graph is of order 𝑥+𝑦+𝑧1 and size 𝑥+𝑦+𝑧 (see [2]). We will denote 𝐾2 as 𝐶2 for convenience.

Lemma 2.4. (i)𝑃𝐶𝑛=(𝜆1)𝑛+(1)𝑛(𝜆1),𝑛2,(2.3)(ii)𝑃𝜃𝑥,𝑦,𝑧=𝑃𝐶𝑥+1𝑃𝐶𝑦+1𝑃𝐶𝑧+1𝜆2(𝜆1)2+𝑃𝐶𝑥𝑃𝐶𝑦𝑃𝐶𝑧𝜆2,if𝑃𝐶𝑧1,𝑥+1𝑃𝐶𝑦+1𝜆(𝜆1)if𝑧=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 𝑥, 𝑦, 𝑧, 𝑛, and 𝜆, let us write 𝑄𝑛(𝜆)=𝑛2𝑖=0(1)𝑖(𝜆1)𝑛2𝑖,𝑀𝑥,𝑦,𝑧(𝜆)=𝑄𝑥+1(𝜆)𝑄𝑦+1(𝜆)𝑄𝑧+1(𝜆)+(𝜆1)2𝑄𝑥(𝜆)𝑄𝑦(𝜆)𝑄𝑧(𝜆).(2.5) Note that when 𝜆=1, we have 𝑄𝑛(1)=(1)𝑛 and 𝑀𝑥,𝑦,𝑧(1)=(1)𝑥+𝑦+𝑧+1. Lemma 2.4 can then be written as the following lemma.

Lemma 2.5 (see [4]). (i)𝑃(𝐶𝑛)=𝜆(𝜆1)𝑄𝑛(𝜆) and (ii)𝑃(𝜃𝑥,𝑦,𝑧)=𝜆(𝜆1)𝑀𝑥,𝑦,𝑧(𝜆).

We also need the following lemma.

Lemma 2.6 (Whitehead and Zhao [12]). A graph 𝐺 contains a cut-vertex if and only if (𝜆1)2𝑃(𝐺).

Lemma 2.6 also implies that if 𝐻𝐺, then 𝐻 is 2-connected if and only if 𝐺 is so.

3. Classification of Graphs

Let be the 𝜒𝑟-closed family of 2-connected (𝑛,𝑛+4)-graphs with three triangles and at least two induced 𝐶4s. In [8], we classified all the 31 types of graph 𝐹 as shown in Figure 1. Since the approach used to classify all the graphs 𝐹 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.

We are now ready to determine the chromaticity of all 31 types of 𝜒𝑟-closed family of 2-connected (𝑛,𝑛+4)-graphs having exactly 3 triangles and at least two induced 𝐶4s as shown in Figure 1. We first note that if 𝐻𝐹𝑖(1𝑖31) in Figure 1, then 𝐻 must be of type 𝐹𝑗(1𝑗31) in Figure 1 as well. For convenience, we will say that the graph 𝐹𝑖, or any of its relatives, is of type (𝑖).

In what follows, we will use 𝐹𝑖(𝛼), instead of 𝐹𝑖, to denote a graph of type (𝑖) that has a path of length 𝛼. We now present our main results in the following theorem.

Theorem 3.1. (1)𝐻𝐹1 if and only if 𝐻 is of type 𝐹1.
(2)𝐻𝐹2(𝑎) if and only if 𝐻 is of type 𝐹2(𝑎).
(3)𝐻𝐹3 if and only if 𝐻 is of type 𝐹3.
(4)𝐻𝐹4 if and only if 𝐻 is of type 𝐹4.
(5)𝐻𝐹5 if and only if 𝐻 is of type 𝐹5.
(6)𝐻𝐹6 if and only if 𝐻𝐹6,𝐹25 or 𝐻 is of type 𝐹22(3).
(7)𝐻𝐹7 if and only if 𝐻𝐹7,𝐹21,𝐹27 or 𝐻 is of type 𝐹31.
(8)𝐹8(𝑏)={𝐹8(𝑏),𝐹28(𝑏)}.
(9)𝐹9 is 𝜒-unique.
(10)𝐹10={𝐹10,𝐹29}.
(11)𝐻𝐹11 if and only if 𝐻 is of type 𝐹11, 𝐹13(3), or 𝐹24.
(12)𝐻𝐹12 if and only if 𝐻 is of type 𝐹12.
(13)𝐻𝐹13(𝑐) if and only if 𝐻 is of type 𝐹13(𝑐) for 𝑐4, and 𝐻𝐹13(3) if and only if 𝐻 is of type 𝐹11,𝐹13(3), or 𝐹24.
(14)𝐻𝐹14 if and only if 𝐻 is of type 𝐹14 or 𝐹18(3).
(15)𝐹15(𝑑) is 𝜒-unique for 𝑑3, and 𝐹15(2)={𝐹15(2),𝐹31}.
(16)𝐹16 is 𝜒-unique.
(17)𝐹17 is 𝜒-unique.
(18)𝐻𝐹18(𝑒) if and only if 𝐻 is of type 𝐹18(𝑒) for 𝑒4, and 𝐻𝐹18(3) if and only if 𝐻 is of type 𝐹14 or 𝐹18(3).
(19)𝐻𝐹19 if and only if 𝐻 is of type 𝐹19.
(20)𝐹20(𝑓)={𝐹20(𝑓),𝐹26(𝑓)}.
(21)𝐻𝐹21 if and only if 𝐻𝐹7, 𝐹21, 𝐹27 or 𝐻 is of type 𝐹31.
(22)𝐻𝐹22() if and only if 𝐻 is of type 𝐹22() for 4, and 𝐻𝐹22(3) if and only if 𝐻𝐹6,𝐹25 or 𝐻 is of type 𝐹22(3).
(23)𝐻𝐹23 if and only if 𝐻 is of type 𝐹23.
(24)𝐻𝐹24 if and only if 𝐻 is of type 𝐹11, 𝐹13(3), or 𝐹24.
(25)𝐻𝐹25 if and only if 𝐻𝐹6,𝐹25 or 𝐻 is of type 𝐹22(3).
(26)𝐹26(𝑗)={𝐹20(𝑗),𝐹26(𝑗)}.
(27)𝐻𝐹27 if and only if 𝐻𝐹7, 𝐹21, 𝐹27 or 𝐻 is of type 𝐹31.
(28)𝐹28(𝑘)={𝐹8(𝑘),𝐹28(𝑘)}.
(29)𝐹29={𝐹10,𝐹29}.
(30)𝐹30={𝐹15(2),𝐹30}.
(31)𝐻𝐹31 if and only if 𝐻𝐹7, 𝐹21, 𝐹27 or 𝐻 is of type 𝐹31.

4. Chromatic Polynomials of the Graphs

Before proving our main result, we present here some useful information about the chromatic polynomial of 𝐹𝑖 (1𝑖31). Let 𝑊(𝑛,𝑘) denote the graph of order 𝑛 obtained from a wheel 𝑊𝑛 by deleting all but 𝑘 consecutive spokes. Also let 𝑊𝑚(5,3) denote the graph obtained from 𝑊(5,3) by identifying the end-vertices of a path 𝑃𝑚 to two non-adjacent degree 3 vertices of 𝑊(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 as shown in the following lemma.

Lemma 4.1. (1)𝑃𝐹1=𝜆(𝜆1)𝑁1(𝜆),(4.1) where 𝑁1(𝜆)=(𝜆2)(𝜆23𝜆+3)(𝜆36𝜆2+13𝜆11) and 𝑁1(1)=3.
(2)𝑃𝐹2=𝐶(𝑎)(𝜆2)𝑃𝑎+1𝑃(𝑊(5,3))𝜆(𝜆1)=𝜆(𝜆1)(𝜆2)2𝜆2𝑄4𝜆+5𝑎+1(𝜆)=𝜆(𝜆1)𝑁2(𝜆),(4.2) where 𝑁2(𝜆)=(𝜆2)2(𝜆24𝜆+5)𝑄𝑎+1(𝜆) and 𝑁2(1)=(1)2(14+5)(1)𝑎+1=2(1)𝑎+1.
(3)𝑃𝐹3=𝜆(𝜆1)𝑁3(𝜆),(4.3) where 𝑁3(𝜆)=(𝜆2)2(𝜆24𝜆+5)(𝜆23𝜆+3) and 𝑁3(1)=2.
(4)𝑃𝐹4=𝜆(𝜆1)𝑁4(𝜆),(4.4) where 𝑁4(𝜆)=(𝜆2)3(𝜆23𝜆+3)2 and 𝑁4(1)=1.
(5)𝑃𝐹5=𝜆(𝜆1)𝑁5(𝜆),(4.5) where 𝑁5(𝜆)=(𝜆2)3(𝜆35𝜆2+10𝜆7) and 𝑁5(1)=1.
(6)𝑃𝐹6=𝜆(𝜆1)𝑁6(𝜆),(4.6) where 𝑁6(𝜆)=(𝜆2)(𝜆24𝜆+5)(𝜆35𝜆2+9𝜆7) and 𝑁6(1)=4.
(7)𝑃𝐹7=𝜆(𝜆1)𝑁7(𝜆),(4.7) where 𝑁7(𝜆)=(𝜆2)2(𝜆36𝜆2+14𝜆13) and 𝑁7(1)=(1)2(16+1413)=4.
(8)𝑃𝐹8(𝑏)=(𝜆2)3𝑃𝐶𝑏+2=(𝜆3)𝑃(𝑊(𝑏+3,3))(𝜆2)3𝑃𝐶𝑏+2𝑃𝐶(𝜆2)(𝜆3)𝑏+2𝐶𝑃𝑏+1𝜆=𝜆(𝜆1)(𝜆2)2𝑄5𝜆+7𝑏+2(𝜆)+(𝜆3)𝑄𝑏+1(𝜆)=𝜆(𝜆1)𝑁8(𝜆),(4.8) where 𝑁8(𝜆)=(𝜆2)[(𝜆25𝜆+7)𝑄𝑏+2(𝜆)+(𝜆3)𝑄𝑏+1(𝜆)] and 𝑁8(1)=(1)[3(1)𝑏+2+(2)(1)𝑏+1]=5(1)𝑏+1.
(9)𝑃𝐹9=𝜆(𝜆1)𝑁9(𝜆),(4.9) where 𝑁9(𝜆)=(𝜆2)2(𝜆36𝜆2+14𝜆14) and 𝑁9(1)=5.
(10)𝑃𝐹10=𝜆(𝜆1)𝑁10(𝜆),(4.10) where 𝑁10(𝜆)=(𝜆2)(𝜆48𝜆3+26𝜆241𝜆+27) and 𝑁10(1)=5.
(11)𝑃𝐹11=𝜆(𝜆1)𝑁11(𝜆),(4.11) where 𝑁11(𝜆)=(𝜆2)2(𝜆47𝜆3+20𝜆228𝜆+17) and 𝑁11(1)=3.
(12)𝑃𝐹12=𝜆(𝜆1)𝑁12(𝜆),(4.12) where 𝑁12(𝜆)=(𝜆2)3(𝜆24𝜆+6) and 𝑁12(1)=3.
(13)𝑃𝐹13(𝑐)=(𝜆2)(𝜆2)2𝑃𝐶𝑐+2𝑃𝐾4𝑃𝐶𝑐+1=𝜆(𝜆1)(𝜆2)3𝑃𝐶𝑐+2(𝜆2)2𝐶(𝜆3)𝑃𝑐+1=𝜆(𝜆1)(𝜆2)2(𝜆2)𝑄𝑐+2(𝜆)(𝜆3)𝑄𝑐+1(𝜆)=𝜆(𝜆1)𝑁13(𝜆),(4.13) where 𝑁13(𝜆)=(𝜆2)2[(𝜆2)𝑄𝑐+2(𝜆)(𝜆3)𝑄𝑐+1(𝜆)] and 𝑁13(1)=(1)2[(1)(1)𝑐+2(2)(1)𝑐+1]=3(1)𝑐+1.
(14)𝑃𝐹14=𝜆(𝜆1)𝑁14(𝜆),(4.14) where 𝑁14(𝜆)=(𝜆2)4(𝜆23𝜆+4) and 𝑁14(1)=2.
(15)𝑃𝐹15=(𝑑)(𝜆2)𝑃(𝑊(𝑑+4,3))(𝜆3)𝑃(𝑊(𝑑+3,3))=(𝜆2)2𝑃𝐶𝑑+3𝐶𝑃𝑑+2𝑃𝐶(𝜆2)(𝜆3)𝑑+2𝐶𝑃𝑑+1=𝜆(𝜆1)(𝜆2)(𝜆2)𝑄𝑑+3(𝜆)(2𝜆5)𝑄𝑑+2(𝜆)+(𝜆3)𝑄𝑑+1(𝜆)=𝜆(𝜆1)𝑁15(𝜆),(4.15) where 𝑁15(𝜆)=(𝜆2)[(𝜆2)𝑄𝑑+3(𝜆)(2𝜆5)𝑄𝑑+2(𝜆)+(𝜆3)𝑄𝑑+1(𝜆)] and 𝑁15(1)=(1)[(1)(1)𝑑+3(3)(1)𝑑+2+(2)(1)𝑑+1]=6(1)𝑑+1.
(16)𝑃𝐹16=𝜆(𝜆1)𝑁16(𝜆),(4.16) where 𝑁16(𝜆)=(𝜆2)(𝜆37𝜆2+19𝜆19) and 𝑁16(1)=6.
(17)𝑃𝐹17=𝜆(𝜆1)𝑁17(𝜆),(4.17) where 𝑁17(𝜆)=(𝜆2)(𝜆48𝜆3+26𝜆241𝜆+25) and 𝑁17(1)=3.
(18)𝑃𝐹18=𝐶(𝑒)(𝜆2)(𝜆1)𝑃(𝑊(𝑒+3,3))(𝜆2)(𝜆3)𝑃𝑒+1=(𝜆1)(𝜆2)2𝑃𝐶𝑒+2𝐶𝑃𝑒+1(𝜆2)2𝐶(𝜆3)𝑃𝑒+1=(𝜆1)(𝜆2)2𝑃𝐶𝑒+22(𝜆2)3𝑃𝐶𝑒+1=𝜆(𝜆1)(𝜆2)2(𝜆1)𝑄𝑒+2(𝜆)2(𝜆2)𝑄𝑒+1(𝜆)=𝜆(𝜆1)𝑁18(𝜆),(4.18) where 𝑁18(𝜆)=(𝜆2)2[(𝜆1)𝑄𝑒+2(𝜆)2(𝜆2)𝑄𝑒+1(𝜆)] and 𝑁18(1)=(1)2[02(1)(1)𝑒+1]=2(1)𝑒+1.
(19)𝑃𝐹19=𝜆(𝜆1)𝑁19(𝜆),(4.19) where 𝑁19(𝜆)=(𝜆2)2(𝜆36𝜆2+14𝜆11) and 𝑁19(1)=2.
(20)𝑃𝐹20𝑊(𝑓)=𝑃𝑓+1𝑃𝐶(5,3)(𝑊(5,3))𝑃𝑓+1𝜃𝜆(𝜆1)=(𝜆2)𝑃𝑓+1,2,2(𝜆2)2𝑃𝐶𝑓+2𝜆(𝜆2)2𝑃𝐶4𝜆+5𝑓+1=𝜆(𝜆1)(𝜆2)𝑀𝑓+1,2,2(𝜆)(𝜆2)2𝑄𝑓+2𝜆(𝜆)(𝜆2)2𝑄4𝜆+5𝑓+1(𝜆)=𝜆(𝜆1)𝑁20(𝜆),(4.20) where 𝑁20(𝜆)=(𝜆2)𝑀𝑓+1,2,2(𝜆)(𝜆2)2𝑄𝑓+2(𝜆)(𝜆2)(𝜆24𝜆+5)𝑄𝑓+1(𝜆) and 𝑁20(1)=(1)(1)𝑓(1)𝑓(1)(2)(1)𝑓+1=4(1)𝑓+1.
(21)𝑃𝐹21=𝜆(𝜆1)𝑁21(𝜆),(4.21) where 𝑁21(𝜆)=(𝜆2)2(𝜆36𝜆2+14𝜆13) and 𝑁21(1)=4.
(22)𝑃𝐹22=()𝑃(𝑊(+3,3))𝑃(𝑊(5,3))𝑃𝐾3𝜆=(𝜆2)2𝑃𝐶4𝜆+5+2𝐶𝑃+1𝜆=𝜆(𝜆1)(𝜆2)2𝑄4𝜆+5+2(𝜆)𝑄+1(𝜆)=𝜆(𝜆1)𝑁22(𝜆),(4.22) where 𝑁22(𝜆)=(𝜆2)(𝜆24𝜆+5)[𝑃(𝑄+2(𝜆)𝑃(𝑄+1(𝜆)] and 𝑁22(1)=(1)(2)[(1)+2(1)+1]=4(1)+1.
(23)𝑃𝐹23=𝜆(𝜆1)𝑁23(𝜆),(4.23) where 𝑁23(𝜆)=(𝜆2)(𝜆24𝜆+5)2 and 𝑁23(1)=4.
(24)𝑃𝐹24=𝜆(𝜆1)𝑁24(𝜆),(4.24) where 𝑁24(𝜆)=(𝜆2)2(𝜆47𝜆3+20𝜆228𝜆+17) and 𝑁24(1)=3.
(25)𝑃𝐹25=𝜆(𝜆1)𝑁25(𝜆),(4.25) where 𝑁25(𝜆)=(𝜆2)(𝜆24𝜆+5)(𝜆35𝜆2+9𝜆7) and 𝑁25(1)=4.
(26)𝑃𝐹26𝑊(𝑗)=𝑃𝑗+1𝑃𝐶(5,3)(𝑊(5,3))𝑃𝑗+1𝜃𝜆(𝜆1)=(𝜆2)𝑃𝑗+1,2,2(𝜆2)2𝑃𝐶𝑗+2𝜆(𝜆2)2𝑃𝐶4𝜆+5𝑗+1=𝜆(𝜆1)(𝜆2)𝑀𝑗+1,2,2(𝜆)(𝜆2)2𝑄𝑗+2𝜆(𝜆)(𝜆2)2𝑄4𝜆+5𝑗+1(𝜆)=𝜆(𝜆1)𝑁26(𝜆),(4.26) where 𝑁26(𝜆)=(𝜆2)𝑀𝑗+1,2,2(𝜆)(𝜆2)2𝑄𝑗+2(𝜆)(𝜆2)(𝜆24𝜆+5)𝑄𝑗+1(𝜆) and 𝑁26(1)=(1)(1)𝑗(1)𝑗(1)(2)(1)𝑗+1=4(1)𝑗+1.
(27)𝑃𝐹27=𝜆(𝜆1)𝑁27(𝜆)(4.27) where 𝑁27(𝜆)=(𝜆2)2(𝜆36𝜆2+14𝜆13) and 𝑁27(1)=4.
(28)𝑃𝐹28(𝑘)=(𝜆2)3𝑃𝐶𝑘+2=(𝜆3)𝑃(𝑊(𝑘+3,3))(𝜆2)3𝑃𝐶𝑘+2𝑃𝐶(𝜆2)(𝜆3)𝑘+2𝐶𝑃𝑘+1𝜆=𝜆(𝜆1)(𝜆2)2𝑄5𝜆+7𝑘+2(𝜆)+(𝜆3)𝑄𝑘+1(𝜆)=𝜆(𝜆1)𝑁28(𝜆),(4.28) where 𝑁28(𝜆)=(𝜆2)[(𝜆25𝜆+7)𝑄𝑘+2(𝜆)+(𝜆3)𝑄𝑘+1(𝜆)] and 𝑁28(1)=(1)[3(1)𝑘+2+(2)(1)𝑘+1]=5(1)𝑘+1.
(29)𝑃𝐹29=𝜆(𝜆1)𝑁29(𝜆),(4.29) where 𝑁29(𝜆)=(𝜆2)(𝜆48𝜆3+26𝜆241𝜆+27) and 𝑁29(1)=5.
(30)𝑃𝐹30=𝜆(𝜆1)𝑁30(𝜆),(4.30) where 𝑁30(𝜆)=(𝜆2)(𝜆48𝜆3+26𝜆242𝜆+29) and 𝑁30(1)=6.
(31)𝑃𝐹31=𝜆(𝜆1)𝑁31(𝜆),(4.31) where 𝑁31(𝜆)=(𝜆2)2(𝜆36𝜆2+14𝜆13) and 𝑁31(1)=4.

Lemma 4.2. Let 1={𝐹4,𝐹5},2={𝐹2,𝐹3,𝐹14,𝐹18,𝐹19}, 3={𝐹1,𝐹11,𝐹12,𝐹13,𝐹17,𝐹24}, 4={𝐹6,𝐹7,𝐹20,𝐹21,𝐹22,𝐹23,𝐹25, 𝐹26,𝐹27,𝐹31}, 5={𝐹8,𝐹9,𝐹10,𝐹28,𝐹29}, and 6={𝐹15,𝐹16,𝐹30}. Then, for each 𝐹𝑖, 𝑖=1,2,3,4,5,6, 𝐻𝐹 implies that 𝐻 must be of type 𝐹 or 𝐹 for an 𝐹 in 𝑖.

Proof. It follows directly from Lemma 4.1 that if 𝑖𝑗, 𝐹𝑝𝑖 and 𝐹𝑞𝑗, then |𝑁𝑝(1)|=𝑖𝑗=|𝑁𝑞(1)|.

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

Lemma 4.3. (1)𝐹6𝐹25.
(2)𝐹7𝐹21𝐹27𝐹31.
(3)𝐹8(𝑏)𝐹28(𝑘) if and only if 𝑏=𝑘.
(4)𝐹10𝐹29.
(5)𝐹11𝐹24.
(6)𝐹20(𝑓)𝐹26(𝑗) if and only if 𝑓=𝑗.

5. Proof of the Main Theorem

We are now ready to prove our main theorem.

(1) Let 𝐻𝐹1. By Lemma 4.2, 𝐻 is of type (1), (11), (12), (13), (17), or (24). If 𝐻=𝐹1, then 𝐻 is of type 𝐹1. Lemma 4.1 further implies that 𝑃(𝐹1,𝜆)𝑃(𝐹𝑖,𝜆), 𝑖=11,12,17,24. Hence, 𝐻 cannot be of type (11), (12), (17), or (24). If 𝐻=𝐹13(𝑐), by Lemma 2.3, 𝑐=3. Using Software Maple, we have 𝑃𝐹13(3)=𝜆(𝜆1)(𝜆2)2𝜆47𝜆3+20𝜆2𝜆28𝜆+17(𝜆2)2𝜆3𝜆+336𝜆2𝐹+13𝜆11=𝑃1.(5.1) Thus, 𝐻 must be of type 𝐹1.

(2) Let 𝐻𝐹2. By Lemma 4.2, 𝐻 is of type (2), (3), (14), (18), or (19). If 𝐻=𝐹2(𝑎), then by Lemma 2.3, 𝑎=𝑎. Thus, 𝐻 must be of type 𝐹2. Since 𝐹2(𝑎) has two induced 𝐶4s while each of 𝐹3 and 𝐹19 has at least three induced 𝐶4s, by Lemma 2.3, 𝐻 cannot be of type (3) or (19). Since 𝑃(𝐹14) is divisible by (𝜆2)4 but not 𝑃(𝐹2(𝑎)), 𝐻 cannot be of type (14). If 𝐻=𝐹18(𝑒), then by Lemma 2.3, 𝑒=𝑎. Note that 𝑃𝐹2(𝑎)=(𝜆1)(𝜆2)3𝑃𝐶𝑎+1(𝜆2)2𝐶(𝜆3)𝑃𝑎+1,𝑃𝐹18=(𝑎)(𝜆1)(𝜆2)𝑃(𝑊(𝑎+3,3))(𝜆2)2𝐶(𝜆3)𝑃𝑎+1.(5.2) This implies that (𝜆2)2𝑃(𝐶𝑎+1)=𝑃(𝑊(𝑎+3,3)), a contradiction since 𝑃(𝑊(𝑎+3,3)) is not divisible by (𝜆2)2. Thus, 𝐻𝐹2(𝑎) if and only if 𝐻 is of type 𝐹2(𝑎).

(3) Let 𝐻𝐹3. By Lemma 4.2 and the above result, 𝐻 is of type (3), (14), (18), or (19). If 𝐻=𝐹3, then 𝐻 is of type 𝐹3. By Lemma 4.1, 𝐹3̸𝐹14 and 𝐹19. If 𝐻=𝐹18(𝑒), by Lemma 2.3, 𝑒=3. Using Software Maple, we have 𝑃𝐹18(3),𝜆=𝜆(𝜆1)(𝜆2)4𝜆23𝜆+4(𝜆2)2𝜆2𝜆4𝜆+52𝐹3𝜆+3=𝑃3.,𝜆(5.3) Thus, 𝐻 must be of type 𝐹3.

(4) Let 𝐻𝐹4. By Lemma 4.2, 𝐻 is of type (4) or (5). If follows directly from Lemma 4.1 that 𝐹4̸𝐹5. Thus, 𝐻 must be of type 𝐹4.

(5) Let 𝐻𝐹5. By Lemma 4.2 and the above result, 𝐻 must be of type (5). Thus, 𝐻 must be of type 𝐹5.

(6) By Lemma 4.2, 𝐻 is of type (6), (7), (20), (21), (22), (23), (25), (26), (27), or (31). If 𝐻=𝐹6, then 𝐻𝐹6. Note that Lemma 4.1 implies that 𝐹6̸𝐹𝑖, 𝑖=7,21,23,27,31. If 𝐻=𝐹20(𝑓), 𝐹22(), or 𝐹26(𝑗), by Lemma 2.3, 𝑓==𝑗=3. Using Software Maple, we have 𝑃𝐹20𝐹(3),𝜆=𝑃26(3),𝜆=𝜆(𝜆1)(𝜆2)2𝜆47𝜆3+20𝜆2𝜆28𝜆+18𝜆(𝜆1)(𝜆2)2𝜆4𝜆+535𝜆2𝐹+9𝜆7=𝑃22𝐹(3),𝜆=𝑃6.,𝜆(5.4) Thus, by Lemma 4.3, 𝐻𝐹6 if and only if 𝐻𝐹6,𝐹25 or of type 𝐹22(3).

(7) Let 𝐻𝐹7. By Lemma 4.2 and the above results, 𝐻 is of type (7), (20), (21), (22) where 4, (23), (26), (27), or (31). If 𝐻=𝐹𝑖, 𝑖=7,21,27,31, Lemma 4.3 implies that 𝐻𝐹7, 𝐹21, 𝐹27, or 𝐻 is of type 𝐹31. Lemma 4.1 further implies that 𝐻 cannot be of type (20), (22), (23), or (26). Thus, 𝐻𝐹7 if and only if 𝐻𝐹7, 𝐹21, 𝐹27, or 𝐻 is of type 𝐹31.

(8) Let 𝐻𝐹8(𝑏). By Lemma 4.2, 𝐻 is of type (8), (9), (10), (28), or (29). If 𝐻=𝐹8(𝑏), by Lemma 2.3, 𝑏=𝑏. Thus, 𝐻𝐹8(𝑏). Since 𝐹8(𝑏) is of order at least 8 but 𝐹𝑖, 𝑖=9,10,29 is of order 7, by Lemma 2.3, 𝑃(𝐹8(𝑏))𝑃(𝐹𝑖), 𝑖=9,10,29. By Lemma 4.3, 𝑃(𝐹8(𝑏))=𝑃(𝐹28(𝑏)). Hence, 𝐹8(𝑏)={𝐹8(𝑏),𝐹28(𝑏)}.

(9) Let 𝐻𝐹9. By Lemma 4.2 and the above results, 𝐻 is of type (9), (10), or (29). By Lemma 4.1, 𝐹9̸𝐹10,𝐹29. Thus, 𝐻𝐹9 and 𝐹9 is 𝜒-unique.

(10) Let 𝐻𝐹10. By Lemma 4.2 and the above result, 𝐻 is of type (10) or (29). By Lemma 4.3, 𝐹10={𝐹10,𝐹29}.

(11) Let 𝐻𝐹11. By Lemma 4.2 and the above result, 𝐻 is of type (11), (12), (13), (17), or (24). If 𝐻=𝐹11 or 𝐹24, by Lemma 4.3, 𝐻 must be of type 𝐹11 or 𝐹24. Lemma 4.1 further implies that 𝑃(𝐹11,𝜆)𝑃(𝐹12,𝜆) and 𝑃(𝐹17,𝜆). Hence, 𝐻 cannot be of type (12) or (17). If 𝐻=𝐹13(𝑐), Lemma 2.3 implies that 𝑐=3. Using Software Maple, we have 𝑃𝐹13(3),𝜆=𝜆(𝜆1)(𝜆2)2𝜆47𝜆3+20𝜆2𝐹28𝜆+17=𝑃11.,𝜆(5.5) Hence, 𝐻𝐹11 if and only if 𝐻 is of type 𝐹11, 𝐹13(3), or 𝐹24.

(12) Let 𝐻𝐹12. By Lemma 4.2 and the above result, 𝐻 is of type (12), (13) with 𝑐4 or (17). Since 𝐹12 and 𝐹13(𝑐) have different order, Lemma 2.3 implies that 𝐹12̸𝐹13. Lemma 4.1 also implies that 𝐹12̸𝐹17. Thus, 𝐻 must be of type 𝐹12.

(13) Let 𝐻𝐹13(𝑐),𝑐4. By Lemma 4.2 and the above result, 𝐻 is of type (13) with 𝑐4 or (17). If 𝐻=𝐹13(𝑐), then 𝑐=𝑐. Since 𝐹13(𝑐) and 𝐹17 have different order, Lemma 2.3 implies that 𝐹13̸(𝑐)𝐹17. Thus, 𝐻𝐹13(𝑐) if and only if 𝐻 is of type 𝐹13(𝑐) for 𝑐4 and 𝐻𝐹13(3) if and only if 𝐻 is of type 𝐹11, 𝐹13(3), or 𝐹24.

(14) Let 𝐻𝐹14. By Lemma 4.2 and the above result, 𝐻 is of type (14), (18) or (19). If 𝐻=𝐹14, then 𝐻 is of type 𝐹14. If 𝐻=𝐹18(𝑒), by Lemma 2.3, 𝑒=3. Using Software Maple, we have 𝑃𝐹18(3),𝜆=𝜆(𝜆1)(𝜆2)4𝜆2𝐹3𝜆+4=𝑃14.,𝜆(5.6) By Lemma 4.1, we also have 𝐹14̸𝐹19. Hence, 𝐻𝐹14 if and only if 𝐻 is of type 𝐹14 or 𝐹18(3).

(15) Let 𝐻𝐹15(𝑑). By Lemma 4.2, 𝐻 must be of type (15), (16), or (30). If 𝐻=𝐹15(𝑑), by Lemma 2.3, 𝑑=𝑑. Thus, 𝐻𝐹15. Since 𝐹16 has exactly six induced 𝐶4s while 𝐹15(𝑑) has only two induced 𝐶4s, by Lemma 2.3, 𝐻 cannot be of type (16). If 𝐻=𝐹31, by Lemma 2.3, 𝑑=2. Using Software Maple, we have 𝑃𝐹15𝜆(2)=𝜆(𝜆1)(𝜆2)48𝜆3+26𝜆2𝐹42𝜆+29=𝑃30.(5.7) Thus, 𝐹15(2)={𝐹15(2),𝐹30} and 𝐹15(𝑑) is 𝜒-unique for 𝑑3.

(16) Let 𝐻𝐹16. By Lemma 4.2 and the above results, 𝐻𝐹16. Thus, 𝐹16 is 𝜒-unique.

(17) Let 𝐻𝐹17. By Lemma 4.2 and the above results, 𝐻𝐹17. Thus, 𝐹17 is 𝜒-unique.

(18) Let 𝐻𝐹18(𝑒), 𝑒4. By Lemma 4.2 and the above results, 𝐻 must be of of type (18) with 𝑒4, or (19). If 𝐻=𝐹18(𝑒), Lemma 2.3 implies that 𝑒=𝑒. Since 𝐹18(𝑒) and 𝐹19 are of different order, it follows that 𝐻 cannot be of type (19). Thus, 𝐻𝐹18(𝑒) if and only if 𝐻 is of type 𝐹18(𝑒) for 𝑒4, and 𝐻𝐹18(3) if and only if 𝐻 is of type 𝐹14 or 𝐹18(3).

(19) Let 𝐻𝐹19. By Lemma 4.2 and the above results, 𝐻 must be of type 𝐹19.

(20) Let 𝐻𝐹20(𝑓). By Lemma 4.2 and the above results, 𝐻 must be of type (20), (22) where 4, (23) or (26). If 𝐻=𝐹20(𝑓), Lemma 2.3 implies that 𝑓=𝑓. If 𝐻=𝐹22(), Lemma 2.3 implies that =𝑓. Note that 𝑃𝐹20=𝑃𝐹(𝑓)(𝜆1)𝑃(𝑊(𝑓+4,4))(𝜆3)𝑃(𝑊(𝑓+3,3)),22(𝑓)=(𝜆1)(𝜆2)𝑃(𝑊(𝑓+3,3))(𝜆3)𝑃(𝑊(𝑓+3,3)).(5.8) This implies that 𝑃(𝑊(𝑓+4,4))=(𝜆2)𝑃(𝑊(𝑓+3,3)), a contradiction since 𝑃(𝑊(𝑓+4,4)) is not divisible by (𝜆2)2 but (𝜆2)𝑃(𝑊(𝑓+3,3)) is divisible by (𝜆2)2. Since 𝐹20 and 𝐹23 are of different order, Lemma 2.3 further implies that 𝐻 cannot be of type (23). Lemma 4.3 then implies that 𝐹20(𝑓)={𝐹20(𝑓),𝐹26(𝑓)}.

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

(22) Let 𝐻𝐹22(), 4. By Lemma 4.2 and the above result, 𝐻 is of type (22) with 4, or (23). If 𝐻=𝐹22(), Lemma 2.3 implies that =. Since 𝐹22() and 𝐹23 are of different order, Lemma 2.3 further implies that 𝐻 cannot be of type (23). Thus, 𝐻𝐹22() if and only if 𝐻 is of type 𝐹22() for 4, and 𝐻𝐹22(3) if and only if 𝐻𝐹6, 𝐹25 or 𝐻 is of type 𝐹22(3).

(23) Let 𝐻𝐹23. By Lemma 4.2 and the above results, 𝐻 must be of type 𝐹23. Thus, 𝐻𝐹23 if and only if 𝐻 is of type 𝐹23.

(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 (𝑛,𝑛+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 (𝑛,𝑛+4)-graphs with exactly 3 triangles is far from completion although all 23 𝜒𝑟-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 (𝑛,𝑛+4)-graphs with exactly 3 triangles are 𝜒-equivalent. Perhaps, the approach used in the study of the chromaticity of 𝐾4-homeomorphs (see [13]) or a more efficient approach of comparing the chromatic polynomials of graphs can be applied in solving the following problem.

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

Acknowledgment

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