[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-150542-en":3,"doc-seo-150542-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},150542,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","On generalized choice and coloring numbers","A paper studies the relationship between coloring number and choosability under relaxed conditions on color classes, represented by a graph parameter f. Building on Alon’s bound connecting coloring number and choosability, it introduces (f,p)-proper colorings and list variants, defining χf,p(G), χ′f,p(G) and the invariant colf,p(G) via s-islands in induced subgraphs. The work proves a lemma linking χ′f,p(G) to colf,p(G) for connected hereditary parameters.","On generalized choice and coloring numbers ∗  \nZdeněk Dvořák† Jakub Pekárek  \nCharles University  \nPrague  \nCzech Republic  \n{rakdver,[pekarej}@iuuk.mff.cuni.cz](pekarej}@iuuk.mff.cuni.cz)  \nJean-Sébastien Sereni‡  \nCentre National de la Recherche Scientifique  \nICube (CSTB)  \nStrasbourg, France  \n[sereni@kam.mmf.cuni.cz](sereni@kam.mmf.cuni.cz)  \nSubmitted: Feb 27, 2018; Accepted: Mar 4, 2019; Published: Mar 22, 2019  \n􀀍c The authors. Released under the CC BY-ND license (International 4.0) .  \nAbstract  \nA well-known result of Alon shows that the coloring number of a graph is bounded by a function of its choosability. We explore this relationship in a more general setting with relaxed assumptions on color classes, encoded by a graph parameter. Mathematics Subject Classifications: 05C15  \nThere exist countless variations of proper graph colorings, where the constraints on the structure of the color classes are either relaxed, stronger or simply different. In other words, instead of requiring color classes to be independent sets, one can require them to have maximum degree, or tree-width, or component sizes bounded from above by a fixed parameter. This article contributes to an effort toward unifying our understanding of such variants of graph coloring.  \nA coloring of a graph G is proper if adjacent vertices receive distinct colors, and the chromatic number χ (G) of G is the least integer s for which G admits a proper coloring using s different colors. A list assignment for G is a function L that to each vertex assigns a set of colors. It is an s-list assignment if |L(v)| > s for each vertex v ∈ V (G) . An L-coloring is a coloring ϕ of G such that ϕ (v) ∈ L (v) for all v ∈ V (G) . An L-coloring  \n∗ This work falls within the scope of L.I.A. STRUCO.  \n†Supported by project 17-04611S (Ramsey-like aspects of graph coloring) of Czech Science Foundation.‡This work was partially supported by A.N.R. Project STINT and P.H.C. Barrande 40625WH.  \nthe electronic journal of combinatorics 26(1) (2019), \\#P1 .51 1  \nis proper if no two adjacent vertices have the same color. The choosability χ` (G) of Gis the least integer s such that G has a proper L-coloring for every s-list assignment L. The coloring number col (G) of G is the least integer s such that every subgraph of G contains a vertex of degree less than s. A straightforward greedy argument shows that χ` (G) 6 col (G) . While the gap between the chromatic number and the coloring number can be arbitrarily large — there are ∆-regular bipartite graphs for every integer ∆ —Alon [1] proved that the same is not true regarding choosability: the coloring number of a graph is bounded by an exponential function of its choosability, which can be seen as a weak converse of the previous upper bound.  \nOne can equivalently define a proper coloring ϕ as one in which, for every color c, its color class ϕ−1 (c) induces an independent set in G. A number of relaxations of this concept have been studied, requiring instead that the color classes induce subgraphs with bounded maximum degree [5 , 6 , 7 , 11 , 13 , 16 , 18 , 20], bounded maximum component size [2 , 10 , 14] or bounded tree-width [4 , 8], for instance. This suggests the following generalization, proposed by Dvořák and Norin [9] . Let f be a graph parameter, assigning to every graph an element of N ∪ {∞}, such that isomorphic graphs are assigned the same value. For an integer p, a coloring of a graph G is (f, p)-proper if f(G[ϕ−1 (c)]) 6 p for each color c. We can now naturally define χf,p (G) as the least number s of colors in an (f, p)-proper coloring of G and χ`f,p(G) as the least integer s such that G has an (f, p)-proper L-coloring for every s-list assignment L of G, or ∞ if no such integer s exists. For example, if f(G) = ∆(G) then χf,p is the defective chromatic number with defect p as introduced by Cowen, Cowen and Woodall [5] . Regarding defective colorings (sometimes called improper colorings) as well as clustered colorings (which both fall","cbCaigf93XxjgZke","https://ap.wps.com/l/cbCaigf93XxjgZke","pdf",398905,1,14,"English","en",105,"# Abstracted framework and motivation\n# Definitions of (f,p)-proper coloring and list versions\n## χf,p(G) and χ′f,p(G)\n# Generalized coloring number via s-islands\n## colf,p(G) properties\n# Hereditary and connected parameters\n## Lemma 1 and proof sketch","[{\"question\":\"What does the paper generalize about proper graph colorings?\",\"answer\":\"It replaces the usual requirement that each color class is an independent set with relaxed constraints captured by a graph parameter f and a bound p.\"},{\"question\":\"How are (f,p)-proper colorings defined?\",\"answer\":\"A coloring is (f,p)-proper if for every color c, the subgraph induced by vertices using c satisfies f(G[ϕ^{-1}(c)]) ≤ p.\"},{\"question\":\"What is the role of s-islands and colf,p(G)?\",\"answer\":\"For each induced subgraph H, colf,p(G) is the smallest s such that H has an s-island I with f(H[I]) ≤ p, providing an analogue of the coloring number relevant to these generalized settings.\"}]","On generalized choice and coloring numbers | 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does the paper generalize about proper graph colorings?","Question",{"text":75,"@type":76},"It replaces the usual requirement that each color class is an independent set with relaxed constraints captured by a graph parameter f and a bound p.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are (f,p)-proper colorings defined?",{"text":80,"@type":76},"A coloring is (f,p)-proper if for every color c, the subgraph induced by vertices using c satisfies f(G[ϕ^{-1}(c)]) ≤ p.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the role of s-islands and colf,p(G)?",{"text":84,"@type":76},"For each induced subgraph H, colf,p(G) is the smallest s such that H has an s-island I with f(H[I]) ≤ p, providing an analogue of the coloring number relevant to these generalized 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