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λ-backbone colorings along pairwise disjoint stars and matchings
Broersma H.J.a, Fujisawa J.b, Marchal L.c, Paulusma D.a, Salman A.N.M.d, Yoshimoto K.b
a Department of Computer Science, Durham University, United Kingdom
b Department of Computer Science, Nihon University, Japan
c Quantitative Economics, Maastricht University, Netherlands
d Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia
e Department of Mathematics, College of Science and Technology, Nihon University, Japan
[vc_row][vc_column][vc_row_inner][vc_column_inner][vc_separator css=”.vc_custom_1624529070653{padding-top: 30px !important;padding-bottom: 30px !important;}”][/vc_column_inner][/vc_row_inner][vc_row_inner layout=”boxed”][vc_column_inner width=”3/4″ css=”.vc_custom_1624695412187{border-right-width: 1px !important;border-right-color: #dddddd !important;border-right-style: solid !important;border-radius: 1px !important;}”][vc_empty_space][megatron_heading title=”Abstract” size=”size-sm” text_align=”text-left”][vc_column_text]Given an integer λ ≥ 2, a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring of (G, H) is a proper vertex coloring V → {1, 2, …} of G, in which the colors assigned to adjacent vertices in H differ by at least λ. We study the case where the backbone is either a collection of pairwise disjoint stars or a matching. We show that for a star backbone S of G the minimum number ℓ for which a λ-backbone coloring of (G, S) with colors in {1, …, ℓ} exists can roughly differ by a multiplicative factor of at most 2 – frac(1, λ) from the chromatic number χ (G). For the special case of matching backbones this factor is roughly 2 – frac(2, λ + 1). We also show that the computational complexity of the problem “Given a graph G with a star backbone S, and an integer ℓ, is there a λ-backbone coloring of (G, S) with colors in {1, …, ℓ}?” jumps from polynomially solvable to NP-complete between ℓ = λ + 1 and ℓ = λ + 2 (the case ℓ = λ + 2 is even NP-complete for matchings). We finish the paper by discussing some open problems regarding planar graphs. © 2008 Elsevier B.V. All rights reserved.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Author keywords” size=”size-sm” text_align=”text-left”][vc_column_text]Adjacent vertices,Chromatic number,Graph G,Matching,Matchings,Multiplicative factors,NP Complete,Open problems,Planar graph,Subgraphs,Vertex coloring[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Indexed keywords” size=”size-sm” text_align=”text-left”][vc_column_text]λ-backbone coloring,λ-backbone coloring number,Matching,Star[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Funding details” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”DOI” size=”size-sm” text_align=”text-left”][vc_column_text]https://doi.org/10.1016/j.disc.2008.04.007[/vc_column_text][/vc_column_inner][vc_column_inner width=”1/4″][vc_column_text]Widget Plumx[/vc_column_text][/vc_column_inner][/vc_row_inner][/vc_column][/vc_row][vc_row][vc_column][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][/vc_column][/vc_row]