[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82299-en":3,"doc-seo-82299-105":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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82299,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","A Polynomial-Time Algorithm for Coloring Perfect Graphs Based on Walk Counting","A polynomial-time algorithm for optimally coloring perfect graphs is developed using only graph-theoretic operations. The method decides whether a perfect graph contains a clique of a prescribed size by iteratively counting walks in the graph with weights assigned to edges and nonedges. A uniform weight initialization is updated in each iteration based on walk counts from the previous round. The approach targets maximum stable sets, maximum cliques, and minimum colorings within perfect-graph structure.","A Polynomial-Time Algorithm for Coloring Perfect Graphs  \nBased on Walk Counting  \nAmir Ali Ahmadi∗, Pravesh K. Kothari†, Yukai Tang∗  \nJuly 13, 2026  \narXiv :2607 .09309v 1 [ cs .DS] 10 Jul 2026  \nAbstract  \nWe present a polynomial-time algorithm for optimally coloring perfect graphs that is based entirely on graph-theoretic operations. At its core, the algorithm decides whether a perfect graph contains a clique of a given size by iteratively counting walks in the graph with certain weights assigned to its edges and nonedges. These weights are initialized according to a uniform scheme and then updated in each iteration based on the walk counts from the previous iteration.  \n1 Introduction  \nIn a simple graph G (V, E), with vertex set V and edge set E, a stable set is a set of pairwise nonadjacent vertices and a clique is a set of pairwise adjacent vertices. A coloring of G is an assignment of colors to the vertices of G such that no two adjacent vertices share the same color. This can also be viewed as partitioning V into stable sets. We denote the size of a maximum stable set of G by α (G) , the size of a maximum clique of G by ω(G), and the minimum number of colors needed to color G by χ (G) .  \nIn this paper, we are concerned with finding a maximum stable set, a maximum clique, and a minimum coloring of a perfect graph. A graph G is called perfect if for every induced subgraph 1 H of G, the chromatic number χ(H) is equal to the clique number ω(H) . This is a highly studied family of graphs both in structural graph theory and in combinatorial optimization. One of the most significant results regarding perfect graphs is the strong perfect graph theorem conjectured by Berge [1] and proven many years later by Chudnovsky, Robertson, Seymour, and Thomas [2] . This theorem states that a graph G is perfect if and only if no induced subgraph of G is an odd cycle of length at least five or the complement of one. Among other implications, this theorem has led to a polynomial-time algorithm for recognizing perfect graphs [3] .  \nIt is well known that the problems of finding a maximum stable set, a maximum clique, and a minimum coloring of a general graph are all NP-hard [4] . In an influential paper, Grötschel, Lovász, and Schrijver proved that these problems can be solved in polynomial time for perfect graphs [5] . Their approach, however, relies on the ellipsoid method and is not regarded as combinatorial. Designing a combinatorial polynomial-time algorithm for finding a maximum stable set/clique or a minimum coloring of a perfect graph is a well-known open problem in graph theory; see, e.g., an interview with Lovász [6], an interview with Chudnovsky [7], [8, Chapter 9], [9, Section 12], [10, Section 1], or [11, Section 1] . Over the years, such algorithms have been designed for several subsets of perfect graphs, including interval graphs [12], chordal graphs [13], claw-free perfect graphs [14], bull-free perfect graphs [15], perfect graphs that do not have a balanced skew partition [16], and bounded degree perfect graphs with no prism or hole of length four as induced subgraphs [11] .  \n∗ Princeton University, Operations Research and Financial Engineering. Partially supported by the Princeton AI Lab Seed Grant, the Princeton SEAS Innovation Grant, and a Research Gift in Mathematical Optimization. Email: [aaa@princeton.edu](aaa@princeton.edu) , [yt3846@princeton.edu](yt3846@princeton.edu)  \n†Princeton University, Computer Science. Partially supported by the Princeton AI Lab Seed Grant and the Princeton SEAS Innovation Grant. Email: [kothari@cs.princeton.edu](kothari@cs.princeton.edu)  \n1 A graph H is an induced subgraph of a graph G if V (H) ⊆ V (G) and any two vertices of H are adjacent if and only if they are adjacent in G.  \nIn this paper, we present a new polynomial-time algorithm for finding a maximum stable set, a maximum clique, and a minimum coloring in any perfect graph. At its core, our algorithm decides if a perfect graph","cbCaifbc57jTKBtN","https://ap.wps.com/l/cbCaifbc57jTKBtN","pdf",675087,2,1,20,"English","en",105,"# Introduction\n## Organization and contributions of the paper","[{\"question\":\"What is the main problem this paper addresses for perfect graphs?\",\"answer\":\"It develops a polynomial-time method to compute maximum stable sets, maximum cliques, and minimum colorings in perfect graphs.\"},{\"question\":\"How does the algorithm decide whether a perfect graph contains a clique of a given size?\",\"answer\":\"It iteratively counts walks with weights on edges and nonedges, updating the weights each round using walk counts from the previous iteration.\"},{\"question\":\"What makes the approach relevant to the open question about combinatorial algorithms?\",\"answer\":\"The paper argues that walk counting is a basic combinatorial operation, so the algorithm may be considered combinatorial even though a universally accepted formal 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is the main problem this paper addresses for perfect graphs?","Question",{"text":75,"@type":76},"It develops a polynomial-time method to compute maximum stable sets, maximum cliques, and minimum colorings in perfect graphs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the algorithm decide whether a perfect graph contains a clique of a given size?",{"text":80,"@type":76},"It iteratively counts walks with weights on edges and nonedges, updating the weights each round using walk counts from the previous iteration.",{"name":82,"@type":73,"acceptedAnswer":83},"What makes the approach relevant to the open question about combinatorial algorithms?",{"text":84,"@type":76},"The paper argues that walk counting is a basic combinatorial operation, so the algorithm may be considered combinatorial even though a universally accepted formal definition is 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