T-colorings, divisibility and circular chromatic number

Robert Janczewski; Anna Maria Trzaskowska; Krzysztof Turowski

doi:10.7151/dmgt.2198

T-colorings, divisibility and circular chromatic number

Abstract

Let T be a T-set, i.e., a finite set of nonnegative integers satisfying 0 ∈ T, and G be a graph. In the paper we study relations between the T-edge spans espT (G) and espd⊙T (G), where d is a positive integer and d ⊙ T = {0 ≤ t ≤ d (max T + 1): d |t ⇒ t/d ∈ T} . We show that espd⊙T (G) = d espT (G) − r, where r, 0 ≤ r ≤ d − 1, is an integer that depends on T and G. Next we focus on the case T = {0} and show that espd⊙{0} (G) = ⌈d(χc(G) − 1)⌉, where χc(G) is the circular chromatic number of G. This result allows us to formulate several interesting conclusions that include a new formula for the circular chromatic number χc(G) = 1 + inf espd⊙{0} (G)/d: d ≥ 1 and a proof that the formula for the T-edge span of powers of cycles, stated as conjecture in [Y. Zhao, W. He and R. Cao, The edge span of T-coloring on graph C d n , Appl. Math. Lett. 19 (2006) 647–651], is true.

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Authors (3)

Robert Janczewski dr hab. inż.
Anna Maria Trzaskowska dr inż.
Krzysztof Turowski dr inż.
- Center for Science of Information Purdue University West

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Category:: Articles
Type:: artykuły w czasopismach
Published in:: Discussiones Mathematicae Graph Theory no. 41, pages 441 - 450,
ISSN: 1234-3099
Language:: English
Publication year:: 2021
Bibliographic description:: Janczewski R., Trzaskowska A. M., Turowski K.: T-colorings, divisibility and circular chromatic number// Discussiones Mathematicae Graph Theory -Vol. 41, (2021), s.441-450
DOI:: Digital Object Identifier (open in new tab) 10.7151/dmgt.2198
Verified by:: Gdańsk University of Technology