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Some families of chromatically unique bipartite graphs

Xiebin Chen
1998 Discrete Mathematics  
A sufficient condition is given for a member 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.  ... 
doi:10.1016/s0012-365x(97)00203-3 fatcat:eya4wxetnjc4xflwoluiubdtcu

Classes of chromatically unique graphs

Mieczysław Borowiecki, Ewa Drgas-Burchardt
1993 Discrete Mathematics  
We prove that 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.  ... 
doi:10.1016/0012-365x(93)90142-g fatcat:ezmxl2wvdvbahffah3jjv6r7uq

On the chromatic uniquenes of bipartite graphs

Pablo M Salzberg, Miguel A López, Reinaldo E Giudici
1986 Discrete Mathematics  
We prove the 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.  ... 
doi:10.1016/0012-365x(86)90146-9 fatcat:u3ektsufd5gdtkgugpvgh2csdm

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.  ... 

Chromatic polynomials of complements of bipartite graphs [article]

Adam Bohn
2012 arXiv   pre-print
Bicliques are complements 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  ... 
arXiv:1103.1355v4 fatcat:gjbneyowtzev7fup44aqkhlkju

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).  ... 

On the chromatic uniqueness of certain bipartite graphs

Y.H. Peng
1991 Discrete Mathematics  
., On (1991) 129-140. the 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  ... 
doi:10.1016/0012-365x(91)90320-2 fatcat:6w2hagiivzdo5ot2usrohpvyqe

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]

Mateusz Kamyczura, Mariusz Meszka, Jakub Przybyło
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] .  ... 
arXiv:2203.02040v1 fatcat:rszduwtdjbhhthdvccs2ossegi

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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