{"paper":{"title":"Subcoloring of (Unit) Disk Graphs","arxiv_id":"2506.19452","primary_cat":"cs.DS","submitted_at":"2025-06-24T09:29:40+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"A subcoloring of a graph is a partition of its vertex set into subsets (called colors), each inducing a disjoint union of cliques.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.05,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"It is a natural generalization of the classical proper coloring, in which each color must instead induce an independent set.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.46,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":false,"role":"background","text":"Similarly to proper coloring, we define the subchromatic number of a graph as the minimum integer k such that it admits a subcoloring with k colors, and the corresponding problem k-Subcoloring which asks whether a graph has subchromatic number at most k.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.19,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.22,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.97,"id":"s3","own":true,"role":"contribution","text":"In this paper, we initiate the study of the subcoloring of (unit) disk graphs.","hedge":{"p":0.95,"label":"asserted"},"own_p":0.84,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.02,"method":0.0,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.97,"interpretation":0.0}},{"i":4,"p":0.86,"id":"s4","own":false,"role":"background","text":"One motivation stems from the fact that disk graphs can be seen as a dense generalization of planar graphs where, intuitively, each vertex can be blown into a large clique--much like subcoloring generalizes proper coloring.","hedge":{"p":0.73,"label":"asserted"},"own_p":0.17,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.59,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.86,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.13}},{"i":5,"p":0.85,"id":"s5","own":false,"role":"result","text":"Interestingly, it can be observed that every unit disk graph admits a subcoloring with at most 7 colors.","hedge":{"p":0.9,"label":"asserted"},"own_p":0.45,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.85,"background":0.13,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":6,"p":0.78,"id":"s6","own":true,"role":"result","text":"We first prove that the subchromatic number can be 3-approximated in polynomial-time in unit disk graphs.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.14,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.78,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.22,"interpretation":0.0}},{"i":7,"p":0.45,"id":"s7","own":true,"role":"contribution","text":"We then present several hardness results for special cases of unit disk graphs which somehow prevents the use of classical approaches for improving this result.","hedge":{"p":0.86,"label":"asserted"},"own_p":0.92,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.36,"background":0.0,"limitation":0.14,"future_work":0.0,"contribution":0.45,"interpretation":0.05}},{"i":8,"p":0.96,"id":"s8","own":true,"role":"result","text":"We show in particular that 2-subcoloring remains NP-hard in triangle-free unit disk graphs, as well as in unit disk graphs representable within a strip of bounded height.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.77,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.96,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}},{"i":9,"p":0.74,"id":"s9","own":true,"role":"result","text":"We also solve an open question of Broersma, Fomin, Ne\\v{s}et\\v{r}il, and Woeginger (2002) by proving that 3-Subcoloring remains NP-hard in co-comparability graphs.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.62,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.74,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.26,"interpretation":0.0}},{"i":10,"p":0.94,"id":"s10","own":true,"role":"result","text":"Finally, we prove that every $n$-vertex disk graph admits a subcoloring with at most $O(\\log^3(n))$ colors and present a $O(\\log^2(n))$-approximation algorithm for computing the subchromatic number of such graphs.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.54,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.94,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":11,"p":0.88,"id":"s11","own":true,"role":"method","text":"This is achieved by defining a decomposition and a special type of co-comparability disk graph, called $\\Delta$-disk graphs, which might be of independent interest.","hedge":{"p":0.47,"label":"asserted"},"own_p":0.93,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.88,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.11,"interpretation":0.01}}]},"compiled":{"main_claim":{"p":1.0,"text":"Finally, we prove that every $n$-vertex disk graph admits a subcoloring with at most $O(\\log^3(n))$ colors and present a $O(\\log^2(n))$-approximation algorithm for computing the subchromatic number of such graphs.","location":"abstract s10","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per 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