Chromaticity of the complements of paths and cycles

Du, Qingyan

doi:10.1016/0012-365x(95)00308-j

Cited by 7 publications

(15 citation statements)

References 6 publications

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“…Since Thus, we get that (x + 3) h(P i ), where 2 i 28 and i = 5, 11, 17, 23. Therefore, (x +3) h(C j ), (3 j 14, j = 9) by Lemma 2.4(1), (4) and Lemma 2.6(2), (x +3) h(T 1,1,k ), (2 k 12, k = 7) by Lemma 2.4 (2), (x +3) h(D l ), (4 l 7) by Lemma 2.4(6)-(8), (10) and Lemma 2.6(2), and (x +3) h(T 1,2,t ), (…”

Section: Lemma 27mentioning

confidence: 83%

“…It is easy to see that all the chromatically unique graphs exhibited in [1,2,7] and many of chromatically unique graphs exhibited in [6,8,11] are special cases of this corollary.…”

Section: Corollary 33 For Any Graphs G With (G) > − 4 G Is Chromatmentioning

confidence: 90%

“…[2] and Liu [6]). For any e ∈ E(G), we have h(G) = h(G − e) + h(G * e), where G − e denotes the graph obtained by deleting the edge e from G. …”

Section: Some Lemmasmentioning

confidence: 95%

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The chromatic equivalence classes of the complements of graphs with the minimum real roots of their adjoint polynomials greater than -4

Ma¹,

Ren²

2008

Discrete Mathematics

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Let G be a graph with order n and G its complement. Denote by (G) the minimum real root of the adjoint polynomial of G. Two graphs G and H are chromatically equivalent if and only if G and H are adjointly equivalent. G is chromatically unique if and only if G is adjointly unique. In this paper, we give a method to determine all chromatic equivalence classes of a graph G with (G) > − 4, by using some results on the minimum real roots of the adjoint polynomial of G. Moreover, we obtain a necessary and sufficient condition for those graphs that are chromatically unique.

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Section: Lemma 27mentioning

confidence: 83%

“…It is easy to see that all the chromatically unique graphs exhibited in [1,2,7] and many of chromatically unique graphs exhibited in [6,8,11] are special cases of this corollary.…”

Section: Corollary 33 For Any Graphs G With (G) > − 4 G Is Chromatmentioning

confidence: 90%

See 1 more Smart Citation

The chromatic equivalence classes of the complements of graphs with the minimum real roots of their adjoint polynomials greater than -4

Ma¹,

Ren²

2008

Discrete Mathematics

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“…Many classes of chromatically unique graphs have been found by applying these properties, see [12][13][14][16][17][18]. In particular, Liu and Li proved that if G = i P ni , then K n − E(G) is -unique when P ni is irreducible [13]; In [12], Liu conjectured that P n is -unique if n = 4 and n is even; Du obtained that if G is a 2-regular graph without C 4 as its subgraph or G is k i=1 P ni , where n i is even and n i ≡ 4 (mod 10), then G is -unique [6]. Du also proposed a problem, i.e., whether it is true that K n − E(P m ) is -unique when m is even and m = 4, where n ¿ m.…”

Section: Introductionmentioning

confidence: 99%

On the minimum real roots of the σ-polynomials and chromatic uniqueness of graphs

Zhao

Zhang

et al. 2004

Discrete Mathematics

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Let ÿ(G) denote the minimum real root of the -polynomial of the complement of a graph G and (G) the minimum degree of G. In this paper, we give a characterization of all connected graphs G with ÿ(G) ¿ − 4. Using these results, we establish a su cient and necessary condition for a graph G with p vertices and (G) ¿ p − 3, to be chromatically unique. Many previously known results are generalized. As a byproduct, a problem of Du (Discrete Math. 162 (1996) 109-125) and a conjecture of Liu (Discrete Math. 172 (1997) 85-92) are conÿrmed.

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“…So far, the chromatic polynomials of very few graphs may be computed and closed formulas of which are given. Chromaticity of the complements of paths and cycles were studied in [3] and in [6], the formulas for chromatic polynomials of the complement P n of a path with n vertices and the complement C n of n-cycle C n with n vertices were given.…”

Section: Introductionmentioning

confidence: 99%