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Some families of chromatically unique bipartite graphs

1998
*
Discrete Mathematics
*

A sufficient condition is given for a member

doi:10.1016/s0012-365x(97)00203-3
fatcat:eya4wxetnjc4xflwoluiubdtcu
*of*K-r(p, q) to be x-*unique*and*some**families**of*z-*unique**bipartite**graphs*are obtained. A conjecture is also proposed. ... Let K -r (p, q) denote the*family**of**graphs*obtained from Kp.q by deleting any r distinct edges. In this paper, we study the*chromaticity**of*the*graphs*in K-r(p, q). ... Acknowledgements The author would like to thank the referee for the careful reading*of*the manuscript and helpful suggestions. ...##
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Classes of chromatically unique graphs

1993
*
Discrete Mathematics
*

We prove that

doi:10.1016/0012-365x(93)90142-g
fatcat:ezmxl2wvdvbahffah3jjv6r7uq
*graphs*obtained from complete equibipartite*graphs*by deleting*some*independent sets*of*edges are*chromatically**unique*. ... Drgas-Burchardt, Classes*of**chromatically**unique**graphs*, Discrete Mathematics Ill (1993) 71-75. ... For large*graphs*, it is extremely difficult to prove*chromatic**uniqueness*.*Some*classes*of**chromatically**unique**graphs*are well known. ...##
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On the chromatic uniquenes of bipartite graphs

1986
*
Discrete Mathematics
*

We prove the

doi:10.1016/0012-365x(86)90146-9
fatcat:u3ektsufd5gdtkgugpvgh2csdm
*chromatic**uniqueness**of*the following infinite*families**of**bipartite**graphs*: Km",+k, l~rt3K""",+k, with m~>2 and 0~ 3, where K~", denote the*graph*obtained from K",n by deleting one edge. ... Chao in 1978 that K"""~+ k is*chromatically**unique*for 0~2. P.M. Salzberg et al. Corollary 1. If X is*chromatically*equivalent to Y, and Y is*bipartite*, then X is*bipartite*. ... Servos for their help in the computational aspects*of*this work. ...##
###
Page 5471 of Mathematical Reviews Vol. , Issue 98I
[page]

1998
*
Mathematical Reviews
*

{For the entire collection see MR 98d:05001.}
98i:05071 05CI5
Chen, Xiebin (PRC-ZZTC; Zhanyzhou)

*Some**families**of**chromatically**unique**bipartite**graphs*. ... A sufficient condition is given for a member*of*K~'(p,q) to be y-*unique*and*some**families**of*y-*unique**bipartite**graphs*are obtained. A conjecture is also proposed.” 98i1:05072 0SC15 Dukes, P. ...##
###
Page 1709 of Mathematical Reviews Vol. , Issue 88d
[page]

1988
*
Mathematical Reviews
*

almost all K, homeomorphs are

*chromatically**unique*. ... Let 2 denote a*family**of**graphs*, and let 2’ be the*family**of**graphs*G satisfying the condition that every subcon- traction*of*G (including G itself) is in 2”. ...##
###
Page 7826 of Mathematical Reviews Vol. , Issue 2002K
[page]

2002
*
Mathematical Reviews
*

A sufficient condition for the

*graph*F(a,b,c,d,e, f) to be*chromatically**unique*is obtained, and a*family**of*6-bridge*graphs*F(2,2,2,2,a,6) is proven to be*chromatically**unique*.” 2002k:05094 05C15 Yin, ... The author surveys*some*interesting results and open problems in*chromatic**graph*theory. ...##
###
Page 1336 of Mathematical Reviews Vol. , Issue 85d
[page]

1985
*
Mathematical Reviews
*

The authors also give eleven such

*graphs*with seven points which are*chromatically**unique*.*Some**of*these are members*of*infinite*families**of**chromatically**unique**graphs*. N. Z. ... (YV-SBOL)*Some**chromatically**unique**graphs*with seven points. Ars Combin. 16 (1983), A, 161-172. ...##
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Chromatic polynomials of complements of bipartite graphs
[article]

2012
*
arXiv
*
pre-print

Bicliques are complements

arXiv:1103.1355v4
fatcat:gjbneyowtzev7fup44aqkhlkju
*of**bipartite**graphs*; as such each consists*of*two cliques joined by a number*of*edges. In this paper we study algebraic aspects*of*the*chromatic*polynomials*of*these*graphs*. ... We derive a formula for the*chromatic*polynomial*of*an arbitrary biclique, and use this to give certain conditions under which two*of*the*graphs*have*chromatic*polynomials with the same splitting field ... I am also indebted to a referee for pointing out the connection between the*chromatic*polynomial*of*a biclique and the matchings*of*its complement; his or her insight led to a significant improvement in ...##
###
Page 642 of Mathematical Reviews Vol. , Issue 96b
[page]

1996
*
Mathematical Reviews
*

They also derive unlabelled versions

*of*the*chromatic*polynomials*of*the following well-known*families**of**graphs*: complete*graphs*, chains, circuits, wheels and theta*graphs*. ... A necessary and sufficient condition for an n- partition*graph*to be*chromatically**unique*is also given. These results further extend the set*of**graphs*which are known to be*chromatically**unique*. ...##
###
Page 5921 of Mathematical Reviews Vol. , Issue 92k
[page]

1992
*
Mathematical Reviews
*

On the

*chromatic**uniqueness**of*certain*bipartite**graphs*. Discrete Math. 94 (1991), no. 2, 129-140. ... The*family**of**graphs*obtained by deleting r distinct edges from K(p, q) (the complete*bipartite**graph*in which the two partite sets consist*of*p and q vertices, respectively) is denoted K~'(p,q). ...##
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On the chromatic uniqueness of certain bipartite graphs

1991
*
Discrete Mathematics
*

., On (1991) 129-140. the

doi:10.1016/0012-365x(91)90320-2
fatcat:6w2hagiivzdo5ot2usrohpvyqe
*chromatic**uniqueness**of*certain*bipartite**graphs*, Discrete Mathematics 94 Let K(p, Q), p s q, denote the complete*bipartite**graph*in which the two partite sets consist*of*p and ... We denote by K-'(p, q), the*family**of*all*graphs*obtained by deleting any r distinct edges from K(p, q). ... Complete*bipartite**graphs*with three edges deleted The*family**of**graphs*, ZC3(p, q), obtained by deleting any 3 distinct edges from K(p, q) consists*of*the following six*bipartite**graphs*Klr3(p, q) (1 G ...##
###
Page 476 of Mathematical Reviews Vol. , Issue 83b
[page]

1983
*
Mathematical Reviews
*

The

*family**of*roots*of*the*chromatic*polynomial*of*a*graph*is called the*chromatic*spectrum*of*the*graph*. ... An infinite*family**of*weakly cyclic*graphs*having integral*chromatic*spectrum is constructed in the present article. D. ...##
###
A note on the conflict-free chromatic index
[article]

2022
*
arXiv
*
pre-print

*Graph*Theory, 2022; 99: 349–358] it was recently proved by means

*of*the probabilistic method that χ'_CF(G)≤ C_1log_2Δ+C_2, where C_1>337 and C_2 are constants, whereas there are

*families*

*of*

*graphs*with ... Let G be a

*graph*with maximum degree Δ and without isolated vertices. An edge colouring c

*of*G is conflict-free if the closed neighbourhood

*of*every edge includes a

*uniquely*coloured element. ... In fact investigating χ ′ CF (G) goes down to study

*of*χ CF (G) for the

*family*

*of*line

*graphs*, cf. [3] . ...

##
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Page 3029 of Mathematical Reviews Vol. , Issue 2002E
[page]

2002
*
Mathematical Reviews
*

(MAL-MALA-IM; Kuala Lumpur);
Ho, Chee-Kit (MAL-MALA-IM; Kuala Lumpur)
On the

*chromatic**uniqueness**of*edge-gluing*of*complete*bipartite**graphs*and cycles. ... Stanislav Jendrol’ (SK-SFRK-GA; KoiSice) 2002¢e:05059 05C15 Yin, Zhi-xiang*Chromatic**uniqueness**of*a*family**of*K;-homeomorphs. (English summary) Northeast. Math. J. 17 (2001), no. 2, 138-142. ...##
###
Page 3597 of Mathematical Reviews Vol. , Issue 2001F
[page]

2001
*
Mathematical Reviews
*

(NZ-MASS-FS; Palmerston North)

*Chromatically**unique**bipartite**graphs*with low 3-independent partition numbers. (English summary) Discrete Math. 224 (2000), no. 1-3, 107-124. ... A*graph*G is called 7-*unique*if every*graph*which is*chromatically*equivalent to G is isomorphic to G. ...
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