For a simple graph G, let P(G;λ) be the chromatic polynomial of G. Two graphs G and H are said to be chromatically equivalent, denoted G ~ H if P(G;λ) = P(H;λ). A graph G is said to be chromatically unique, if H ~ G implies that H ≅ G. Chia [4] determined the chromatic equivalence class of the graph consisting of the join of p copies of the path each of length 3. In this paper, we determined the chromatic equivalence class of the graph consisting of the join of p copies of the complete tripartite graph K1,2,3. MSC: 05C15;05C6

Lau, Gee Choon

Peng, Yee Hock

Universiti Putra Malaysia Institutional Repository

Chromatic Equivalence Class of the Join of Certain Tripartite Graphs103Malaysian Journal of Mathematical SciencesChromatic Equivalence Class of the Join of Certain Tripartite Graphs1,3G.C. Lau & 2,3Y.H. Peng1Faculty of I. T. and Quantitative ScienceUniversiti Teknologi MARA (Johor Branch)Segamat, Johor, Malaysia2 Department of Mathematics, and3 Institute for Mathematical ResearchUniversiti Putra Malaysia 43400 UPM Serdang, MalaysiaE-mail: yhpeng@fsas.upm.edu.myMalaysian Journal of Mathematical Sciences 1(1): 103 - 108  (2007)ABSTRACTFor a simple graph G, let P(G;λ) be the chromatic polynomial of G. Two graphs G andH are said to be chromatically equivalent, denoted G ~ H if P(G;λ) = P(H;λ). A graphG is said to be chromatically unique, if  H ~ G implies that H   ≅  G. Chia [4] determinedthe chromatic equivalence class of the graph consisting of the join of p copies of thepath each of length 3. In this paper, we determined the chromatic equivalence class ofthe graph consisting of the join of p copies of the complete tripartite graph K1,2,3.MSC: 05C15;05C60Keywords:  Tripartite graphs; Chromatic polynomial; Chromatic equivalence classINTRODUCTIONAll graphs considered in this paper are finite, undirected, simple and loopless. For agraph G, we denote by P(G;λ) (or P(G)), the chromatic polynomial of G. Two graphs Gand H are said to be chromatically equivalent, or  χ-equivalent, denoted G ~ H if P(G)= P(H). It is clear that the relation " ~ " is an equivalence relation on the family ofgraphs. We denote by [G] the equivalence class determined by G under " ~ ". A graph Gis said to be chromatically unique, or  χ-unique, if [G] = {G}, i.e., H ~ G implies that.GH ≅  Many families of χ-unique graphs are known (see [8, 9]), relatively fewerresults concerning the chromatic equivalence class of graphs are known (see [2, 3,4]). In this paper, our main purpose is to determine the chromatic equivalence class ofthe graph consisting of the join of p copies of the complete tripartite graph K1,2,3.In what follows, we let Kn denote the complete graph on n vertices, ptppK ,...,2,1 thecomplete t-partite graph having ni vertices in the i-th partite set, Pn and Cn the path andcycle on n vertices, respectively and χ(G) the chromatic number of G. Let Wn denotethe wheel of order n and Un the graph obtained from Wn by deleting a spoke of Wn. Alsolet n(A,G) denote the number of subgraph A in G and i(A,G) the number of inducedsubgraph A in G.G.C. Lau & Y.H. Peng104 Malaysian Journal of Mathematical SciencesThe join of two graphs G and H, denoted G + H, is the graph obtained from the unionof G and H by joining every vertex of G to every vertex of H.Let F be a graph and let G = F + F + . . .+ F or pF denote the join of p (≥2) copies ofF. We wish to determine [G]. Let Jp(F) denote the set of all graphs H which are of theform H = H1 + H2 + . . . + Hp, where Hi∈[F], i = 1, 2, . . . ,p.In [4], Chia posed the following problemProblem: What are those graphs F for which Jp(F) = [G]?.and solve the problem for the case F = P4. In this paper, by making very minor modify-cation to the technique used in [4], we solve the above problem for the case F = K1,2,3.PRELIMINARY RESULTS AND NOTATIONSA spanning subgraph is called a clique cover if its connected components are completegraphs. Let G be a graph on n vertices. Let sk(G) denote the number of clique cover of Gwith k connected components, k = 1, 2, . . . ,n. If the chromatic polynomial of G isP(G,λ) = ∑ =nk kk Gs1 )λ)((  where (λ)k = λ (λ  1) · · · (λ  k + 1), then the polynomialσ(G, k) = knk kxGs∑ =1 )( is called the  σ-polynomial of G (see Brenti(1992)). It is easyto see that  σ (G, x) = xn if and only if G = Kn since sk(G) = 0 for k < χ(G) = n. Alsonote that sn(G) = 1 and sn”1(G) = m if G has m edges. Clearly, P(G,λ) = P(H,λ) if andonly if  σ(G, x) = σ(H, x) and sk(G) = sk(H) for k = 1, 2, . . . .If  σ(G, x) = xf(x) for some irreducible polynomial f(x) over the rational number field,then  σ(G, x) is said to be irreducible.Lemma 2.1. (Farrell (1980)) Let G and H be two graphs such that G ~ H. Then G andH have the same number of vertices, edges and triangles. If both G and H has no K4 assubgraph, then i(C4, G) = i(C4, H). Moreover,                     ),(3),(2),(),( 553,25 GWiGUiGKiGCi +++−              ),(3),(2),(),( 553,25 HWiHUiHKiHCi +++−= .Lemma 2.2. (Brenti (1992)) Let G and H be two disjoint graphs. Then),(),(),( xHxGxHG σσσ =+ .In particular,∏ == ti xnOxnnnK it 1 ),(),,...,,( σσ 21 .Chromatic Equivalence Class of the Join of Certain Tripartite Graphs105Malaysian Journal of Mathematical SciencesLemma 2.3. (Liu (1992)) Let G be a connected graph with n vertices and m edges.Assume that G is not the complete graph K3. Then                ⎟⎟⎠⎞⎜⎜⎝⎛ −≤− 21)(2mGsnand equality holds if and only if G is the path Pm+1.A CHROMATIC EQUIVALENCE CLASSWe first have the following lemma which follows readily from Lemma 2.1.Lemma 3.1. [K1,2,3] = {K1,2,3, K2,2,2  e} where e is an edge of .2,2,2K .We now have our main theorem as follow.Theorem 3.1. Let G = K1,2,3 + K1,2,3 + … + K1,2,3 be the join of p copies of K1,2,3. Then[ ] ).( 3,2,1KJG p=Proof. Let H ~ G, we will show that ).( 3,2,1KJH p∈  Since P(G) = P(H) implies that),()( HG σσ =  it is more convenient to look at  σ(G) and σ(H). First note that)()3()( 2,2233,1 eKxxxxK −=++= σσ with [K1,3] = {K1,3, K2,2   e}, and).()3)(()( 2,2,22323,2,1 eKPxxxxxxK −=+++=σ  So,  σ(G) = [x(x2 + x)(x3 +3x2 + x)]p = [(x2 +x)(x4 + 3x3 + x2)]p, having p irreducible factors of x, x2 + x and x3 + 3x2+ x respectively.Let n and m denote the number of vertices and edges in H respectively. Then n = 6pand m = 36 ( )2p  +11p = 18p2 - 7p so that σ(H) =  σ(G) = ipi i xGs∑=6 1 )(   . Moreover, His uniquely 3p-colorable as G is so.Let pVVV 321 ,...,, be the color classes of the unique 3p-coloring of H. Let Vij denote thesubgraph induced by .,1 jiVV j ≠∪  Call Vij a 2-color subgraph of H.Case (i): Every Vi has exactly two vertices.In this case, Vij is either a path P4 or else a cycle C4 because, by Theorem 12.16 of [6], Vijis connected for  i ≠ j.  Note that the number of 2-color subgraphs in H is( ) pppp +−= )59( 22123  . By looking at the number of edges in H, we see that exactly pG.C. Lau & Y.H. Peng106 Malaysian Journal of Mathematical Sciencesof the 2-color subgraphs Vij are 4P  and the rest of the 2-color subgraphs are 4C . Thismeans that H  has only P4 and K2 as subgraph so that 24 KrPsH += (s,r ≥ 0).Consequently,)()]3)(([])()3[()( 2322234 GxxxxxxxxxxxH prs σσ =+++=+++= .Obviously, s, r ≥ 1 so that σ (H) = (x4 + 3x3 + x2)(x2 + x)σ(H1) and that by Lemma 3.1,H = (K2,2,2 − e) + H1 for some graph H1. Since  σ(H1) = [x(x2 + x)(x3 + 3x2 + x)]p-1, byinduction on p, we have H1 ∈ Jp-1(K1,2,3). This implies that H ∈ Jp(K1,2,3).Case (ii): Not every iV  has exactly two vertices.Then there is a j such that |Vj| = 1. Without loss of generality, let |Vj| = i for j = 1, …,r, r  ≥1. Then H = Kr + H*  for some graph H*. Let F1, F2, …, Ft be the connectedcomponents of *H  . Then H = Kr + 1F  +…+ tF  with *H = 1F  +…+ tF  .If for some i, Fi = K3, then H  contains a subgraph K1 ∪ K3. This means that H = K1,3 +H' for some graph H' and so        σ(H) = (x4 + 3x3 + x2)σ(H' )=[x(x2 + x)(x3 + 3x2 + x)]p =σ (G).Clearly, σ(H' ) must contain a factor (x2 + x) so that)()()3()( 1''234 HHxxxH σσσ ++=  (where ))( 2'' xxH +=σ  for some graphH1. Obviously, "H =K2. Hence, H=K1,2,3+H1 with12321 )]3)(([)(−+++= pxxxxxxHσ . Again, by induction on p, we haveH∈Jp(K1,2,3).If for some i, Fi = K2, then H = K2 + H'. By the similar argument as above,  σ(H' ) mustcontain a factor (x3 + 3x2 + x) so that H = K1,2,3 + H1 or (K2,2,2 − e) + H1 with  σ(H1) =[x(x2 + x)(x3 + 3x2 + x)]p-1. Again, by induction on p, we have H∈Jp(K1,2,3).If for some i, Fi = P4 (=K2,2− e), then H = P4 + H'. By the similar argument as above,σ(H') must contain a factor (x2+x) so that H = (K2,2,2 − e) + H1 with  σ (H1) = [x(x2 +x)(x3 + 3x2 + x)]p-1. Again, by induction on p, we have H∈Jp(K1,2,3).So, assume that Fi is not K2, K3 or P4 for any i = 1, …, t. Let ni and mi denote the numberof vertices and edges in Fi respectively. Then  1tiim=∑ = 4p, the number of edges inH .If ni ≤ 3, then Fi = P3. However, this is impossible because  σ(G) does not containChromatic Equivalence Class of the Join of Certain Tripartite Graphs107Malaysian Journal of Mathematical Sciences(x3 + 2x2) as a factor. Hence, ni ≥ 4. This implies that 6p = |V(G)| = r + 1tiin=∑  ≥ r + 4tso that t < 3p/2 because r ≥1.Since H = Kr + H*, we have σ(H) = xr . )( *Hσ  It follows that *2 2 *( ) ( )n ns H s H− −=where n*   is the number of vertices in H* . Note that                       )(*1 1σ)*()*(σ iFnjtijxHjsH ∑= ∏===where          ...,)()()(σ22111 +++== −−−=∑ iiii ninninnkkiki xFsxmxxFsFi = 1, …, t.By multiplying all the terms in ( )1t Fi iσ∏ =   and by equating the coefficient of * 2nx −  , wehave by Lemma 2.3,   )()( 1 21*2* iti ntji jinFsmmHsi∑∑ = −≤≤≤− +=ρ            ∑∑ =≤≤ ⎟⎟⎠⎞⎜⎜⎝⎛ −+≤ tiiji jimmm11 21.Consequently,2)23(2)( 1 12*2*∑ ∑≤≤≤ =−+−+≤ tjiti iijinmmmmHs               ( )21 13 22t ti ii im m t= =− += ∑ ∑   216 12 22p p t− +=                           2916 2 pp −<because t < 3p/2. However, this is a contradiction because ( )2 6 2( )n ps H s G =− −=).(2/)816(2164 *22*Hspppp n −>−=⎟⎟⎠⎞⎜⎜⎝⎛+ This completes the proof.Remark: Note that for even p, our main result is a special case of Theorem 5.1 in (Ho,(2004)).G.C. Lau & Y.H. Peng108 Malaysian Journal of Mathematical SciencesACKNOWLEDGEMENTSThe authors wish to thanks the referees for their valuable comments and suggestions.REFERENCESF. BRENTI, Expansions of chromatic polynomial and log-concavity, Trans. Amer. Math. Soc.332 (1992) 729-756.C.Y. CHAO, On tree of polygons, Arch. Math. 45 (1985) 180-185.G.L. CHIA, On the chromatic equivalence class of a family of graphs, Discrete Math. 162 (1996)285-289.G.L. CHIA, On the chromatic equivalence class of graphs, Discrete Math. 178 (1998) 15-23.E.J. FARRELL, On chromatic coefficients, Discrete Math. 29 (1980) 257-264.F. HARARY, Graph Theory (Addison-Wesley, Reading, MA, 1969).C.K. HO, On graphs determined by their chromatic polynomials, Ph.D. thesis (2004) UniversityMalaya, Malaysia.K.M. KOH AND K.L. TEO, The search for chromatically unique graphs, Graphs Combin. 6(1990) 259-285.K.M. KOH AND K.L. Teo, The search for chromatically unique graphs - II, Discrete Math. 172(1997) 59-78.R.Y. LIU, Chromatic uniqueness of Kn - E(kPs   rPt), J. System Sci. Math. Sci. 12 (1992) 207-214(Chinese, English Summary).

Chromatic equivalence class of the join of certain tripartite graphs

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Chromatic equivalence class of the join of certain tripartite graphs

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