`ISRN Discrete MathematicsVolumeย 2011, Article IDย 430396, 15 pageshttp://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

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

#### 1. Introduction

Let , or simply , denote the chromatic polynomial of a simple graph . Two graphs and are chromatically equivalent (simply -equivalent), denoted , if . A graph is chromatically unique (simply -unique) if implies that . Let denote the equivalence class determined by the graph under ~. Clearly, is -unique if and only if . A graph is called a relative of if there is a sequence of graphs such that each is a -gluing of some graphs (say and ) and that is obtained from by forming another -gluing of and for . We say is a graph of type if is a relative of or . A family of graphs is said to be relative-closed (simply -closed) if(i)no two graphs in are relatives of each other, (ii)for any graph , implies that or is a relative of a graph in .

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

If is a graph of order and size , we say is an -graph. The chromatic equivalence classes of 2-connected -graph have been fully determined for in [1, 2] and partially determined for in [3โ5]. Peng and Lau have also characterized and classified certain chromatic equivalence classes of 2-connected -graph in [6, 7]. In [8], by using the idea of cyclomatic number, the authors obtained the -closed family of 2-connected -graphs with exactly three triangles.

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

#### 2. Notation and Basic Results

Let (or -cycle) be the cycle of order . An induced 4-cycle is the cycle 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 be a graph and an edge of . Then where is the graph obtained from by deleting , and is the graph obtained from by identifying the end vertices of .

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

Lemma 2.2 (Zykov [11]). Let be a -gluing of and . Then Lemma 2.2 implies that all -gluings of and are -equivalent. It follows from Lemma 2.2 that if is a relative of , then .

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

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

A generalized -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 , and , respectively, where . The graph is of order and size (see [2]). We will denote as for convenience.

Lemma 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 , , , , and , let us write Note that when , we have and . Lemma 2.4 can then be written as the following lemma.

Lemma 2.5 (see [4]). and .

We also need the following lemma.

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

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

#### 3. Classification of Graphs

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

Figure 1: 31 types of 2-connected ()-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 -closed family of 2-connected -graphs having exactly 3 triangles and at least two induced s as shown in Figure 1. We first note that if in Figure 1, then must be of type in Figure 1 as well. For convenience, we will say that the graph , or any of its relatives, is of type .

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

Theorem 3.1. if and only if is of type .
if and only if is of type .
if and only if is of type .
if and only if is of type .
if and only if is of type .
if and only if or is of type .
if and only if or is of type .
.
is -unique.
.
if and only if is of type , , or .
if and only if is of type .
if and only if is of type for , and if and only if is of type , or .
if and only if is of type or .
is -unique for , and .
is -unique.
is -unique.
if and only if is of type for , and if and only if is of type or .
if and only if is of type .
.
if and only if , , or is of type .
if and only if is of type for , and if and only if or is of type .
if and only if is of type .
if and only if is of type , , or .
if and only if or is of type .
.
if and only if , , or is of type .
.
.
.
if and only if , , or is of type .

#### 4. Chromatic Polynomials of the Graphs

Before proving our main result, we present here some useful information about the chromatic polynomial of (). Let denote the graph of order obtained from a wheel by deleting all but consecutive spokes. Also let denote the graph obtained from by identifying the end-vertices of a path to two non-adjacent degree 3 vertices of . 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. where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .
where and .

Lemma 4.2. Let ,, , , , , and . Then, for each , , implies that must be of type or for an in .

Proof. It follows directly from Lemma 4.1 that if , and , then .

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

Lemma 4.3. .
.
if and only if .
.
.
if and only if .

#### 5. Proof of the Main Theorem

We are now ready to prove our main theorem.

(1) Let . By Lemma 4.2, is of type (1), (11), (12), (13), (17), or (24). If , then is of type . Lemma 4.1 further implies that , . Hence, cannot be of type (11), (12), (17), or (24). If , by Lemma 2.3, . Using Software Maple, we have Thus, must be of type .

(2) Let . By Lemma 4.2, is of type (2), (3), (14), (18), or (19). If , then by Lemma 2.3, . Thus, must be of type . Since has two induced s while each of and has at least three induced s, by Lemma 2.3, cannot be of type (3) or (19). Since is divisible by but not , cannot be of type (14). If , then by Lemma 2.3, . Note that This implies that , a contradiction since is not divisible by . Thus, if and only if is of type .

(3) Let . By Lemma 4.2 and the above result, is of type (3), (14), (18), or (19). If , then is of type . By Lemma 4.1, and . If , by Lemma 2.3, . Using Software Maple, we have Thus, must be of type .

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

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

(6) By Lemma 4.2, is of type (6), (7), (20), (21), (22), (23), (25), (26), (27), or (31). If , then . Note that Lemma 4.1 implies that , . If , , or , by Lemma 2.3, . Using Software Maple, we have Thus, by Lemma 4.3, if and only if or of type .

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

(8) Let . By Lemma 4.2, is of type (8), (9), (10), (28), or (29). If , by Lemma 2.3, . Thus, . Since is of order at least 8 but , is of order 7, by Lemma 2.3, , . By Lemma 4.3, . Hence, .

(9) Let . By Lemma 4.2 and the above results, is of type (9), (10), or (29). By Lemma 4.1, . Thus, and is -unique.

(10) Let . By Lemma 4.2 and the above result, is of type (10) or (29). By Lemma 4.3, .

(11) Let . By Lemma 4.2 and the above result, is of type (11), (12), (13), (17), or (24). If or , by Lemma 4.3, must be of type or . Lemma 4.1 further implies that and . Hence, cannot be of type (12) or (17). If , Lemma 2.3 implies that . Using Software Maple, we have Hence, if and only if is of type , , or .

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

(13) Let . By Lemma 4.2 and the above result, is of type (13) with or (17). If , then . Since and have different order, Lemma 2.3 implies that . Thus, if and only if is of type for and if and only if is of type , , or .

(14) Let . By Lemma 4.2 and the above result, is of type (14), (18) or (19). If , then is of type . If , by Lemma 2.3, . Using Software Maple, we have By Lemma 4.1, we also have . Hence, if and only if is of type or .

(15) Let . By Lemma 4.2, must be of type (15), (16), or (30). If , by Lemma 2.3, . Thus, . Since has exactly six induced s while has only two induced s, by Lemma 2.3, cannot be of type (16). If , by Lemma 2.3, . Using Software Maple, we have Thus, and is -unique for .

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

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

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

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

(20) Let . By Lemma 4.2 and the above results, must be of type (20), (22) where , (23) or (26). If , Lemma 2.3 implies that . If , Lemma 2.3 implies that . Note that This implies that , a contradiction since is not divisible by but is divisible by . Since and are of different order, Lemma 2.3 further implies that cannot be of type (23). Lemma 4.3 then implies that .

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

(22) Let , . By Lemma 4.2 and the above result, is of type (22) with , or (23). If , Lemma 2.3 implies that . Since and are of different order, Lemma 2.3 further implies that cannot be of type (23). Thus, if and only if is of type for , and if and only if , or is of type .

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

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

Figure 2: Relative-closed family of 2-connected -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 -graphs with exactly 3 triangles.

#### Acknowledgment

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

#### References

1. C. Y. Chao and E. G. Whitehead Jr., โOn chromatic equivalence of graphs,โ in Theory and Applications of Graph, Y. Alavi and D. R. Lick, Eds., vol. 642 of Lecture Notes in Mathematics, pp. 121โ131, Springer, New York, NY, USA, 1978.
2. B. Loerinc, โChromatic uniqueness of the generalized ฮธ-graph,โ Discrete Mathematics, vol. 23, no. 3, pp. 313โ316, 1978.
3. F. M. Dong, K. L. Teo, and K. M. Koh, โA note on the chromaticity of some 2-connected ($n,n+3$)-graphs,โ Discrete Mathematics, vol. 243, no. 1โ3, pp. 217โ221, 2002.
4. K. M. Koh and K. L. Teo, โChromatic classes of 2-connected ($n,n+3$)-graphs with at least two triangles,โ Discrete Mathematics, vol. 127, no. 1โ3, pp. 243โ258, 1994.
5. K. L. Teo and K. M. Koh, โChromatic classes of certain 2-connected ($n,n+2$)-graphs,โ Ars Combinatoria, vol. 32, pp. 65โ76, 1991.
6. Y. H. Peng and G. C. Lau, โChromatic classes of 2-connected ($n,n+4$)-graphs with at least four triangles,โ Discrete Mathematics, vol. 278, no. 1โ3, pp. 209โ218, 2004.
7. Y. H. Peng and G. C. Lau, โChromatic classes of 2-connected ($n,n+4$)-graphs with three triangles and one induced 4-cycle,โ Discrete Mathematics, vol. 309, no. 10, pp. 3092โ3101, 2009.
8. G. C. Lau and Y. H. Peng, โRelative-closed family of graphs with exactly three triangles,โ Technical Report 2, Institute for Mathematical Research, Putra, Malaysia, 2010.
9. M. Behzad, G. Chartrand, and L. Lesniak-Foster, Graphs and Digraphs, Wadsworth, Belmont, Calif, USA, 1979.
10. H. Whitney, โThe coloring of graphs,โ Annals of Mathematics, vol. 33, pp. 688โ718, 1932.
11. A. A. Zykov, โOn some properties of linear complexes,โ Matematicheskii Sbornik, vol. 24, pp. 163โ188, 1949, Translation in American Mathematical Society Translations, no. 79.
12. E. G. Whitehead Jr. and L. C. Zhao, โCutpoints and the chromatic polynomial,โ Journal of Graph Theory, vol. 8, pp. 371โ377, 1984.
13. X. F. Li, โA family of chromatically unique ${K}_{4}$-homeomorphs,โ Journal of Anhui University of Technology and Science, vol. 32, no. 4, pp. 18โ21, 2008.