Abstract
This paper constructively proves the following result: Suppose that k⩾3d−2, (k,d)=1, A is a finite set and f 1,f 2,…,f n are mappings from A to {0,1,…,k−1}. Then, for any integer l, there is a graph G=(V,E) of girth at least l with A⊂V, such that G has exactly n (k,d)-colorings (up to a permutation of the colors) g 1,g 2,…,g n , and each g i is an extension of f i . This result generalizes a result of Müller who proved this for k-colorings. Note that for n=1, the method presented in this paper gives a construction of uniquely (k,d)-colorable graphs.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
