[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84295-en":3,"doc-seo-84295-105":29,"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":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":13,"seo_description":14,"update_tm":27,"read_time":28},84295,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","An Upper Bound on the Hat Guessing Number of Graphs","Hat guessing number HG(G) measures the largest number of hat colors k for which players on vertices can agree on a simultaneous, no-communication strategy guaranteeing that at least one vertex guesses its own hat correctly. The paper proves a general upper bound on HG(G) in terms of the graph order n and maximum degree Δ. The bound is the first nontrivial one in the regime Δ ≥ n/e. It yields asymptotically HG(Gn,1/2) ≤ c n for c ≈ 0.809 and shows graphs with maximum degree (1−ε)n cannot have HG(G) close to n.","arXiv :2607 .07994v1 [math .CO] 8 Jul 2026  \nAN UPPER BOUND ON THE HAT GUESSING NUMBER OF GRAPHS  \nMASON SHURMAN AND SCOTT ALBERT SIBLEY  \nAbstract. The hat guessing number HG (G) of a graph is defined by the following game: each player is placed on a vertex and assigned a hat with one of k colors. Each vertex can see only the hat color of the other vertices it is connected to in G. All vertices guess, simultaneously, the color of their own hat. The hat guessing number HG (G) is the largest k such that the players can guarantee that at least one of them guesses correctly. In this paper, we show a general bound on the hat guessing number of a graph G as a function of its order n and its maximum degree ∆ . This is the first nontrivial upper bound on HG (G) asa function of ∆ and n when ∆ ≥ ~~n~~e . From this result we also obtain that the hat guessing number of the random graph Gn,1/2 is at most asymptotically cn for c ∼ 0.809, and that graphs with maximum degrees of (1 − ε)n for fixed ε > 0 cannot have HG (G) = (1 − o(1))n.  \n1. Introduction  \nConsider the following game: Let G be a graph with n vertices, v 1 , . . . , vn. Each vertex is associated with a player and assigned a color from a set {1,..., k} of k possible colors. Each player may only see the hat colors of their neighbors in G, excluding themselves, so if player i is identified with vi , they can see the hat color of player j identified with vj if and only if vi vj ∈ E (G) . All the players agree beforehand on a guessing strategy, and must simultaneously guess the color of their own hat, without communication. The goal of the players is to guarantee that under any coloring, at least one player guesses correctly, in which case the players win. Let the hat guessing number HG (G) be the largest k such that the players have a winning strategy. The trivial upper bound for the hat guessing number is n, since each player guesses correctly with probability 1/k, where k is the number of colors, so if there were more than n colors, the expected number of correct guesses would be less than 1 .  \nThe hat guessing number has been studied in many papers including [2, 4–9] . One immediate question, asked by Alon, Ben-Eliezer, Shangguan, and Tamo in [2], is whether or not the hat guessing number is bounded above by a function of the maximum degree. As they note, by folklore, a basic application of the Lov´asz Local Lemma yields that if G has maximum degree ∆, HG (G) \u003C e∆ . (Note that if a coloring is chosen randomly, the event that a particular vertex guesses right depends on at most ∆ other vertices.) However, this argument does not completely resolve their question. In the regime where the maximum degree ∆ is large, this bound is worse than the trivial bound of n. Until now, there have not been upper bounds on the hat guessing number as a function of the maximum degree in the regime (∆ > n/e) . Our main result is the first such upper bound.  \nTheorem 1.1 . For any graph G on n vertices with maximum degree ∆,  \n∆ p (∆ + 2)2 + 2(n2 − 3n − n∆)  \nHG (G) ≤ 1 + ~~ ~~ + ~~ ~~ .  \n2 2  \nOne of the most interesting applications of our result is to the random graph Gn,1/2 . Note that the maximum degree of Gn,1/2 is (1 + o(1))n/2 ≥ n/e, so the Lov´asz Local Lemma argument does not yield anontrivial result. In 2019, Bosek et [al. in](al. in) [4] showed that with high probability, that is, with probability tending to 1 as n tends to infinity,  \n(2 + o(1))log2 (n) ≤ HG(Gn,1/2) ≤ n − (1 + o(1))log2 n.  \nIn 2020, Alon and Chizewer in [1] showed that with high probability,  \nHG (Gn,1/2) ≥ n 1−o(1)  \nDate: July 2026 .  \nThe authors would like to thank Asaf Ferber for his comments and mentorship.  \n2 MASON SHURMAN AND SCOTT ALBERT SIBLEY  \nby finding a subgraph with large hat guessing number. Using our main result, we get a better upper bound on HG(Gn,1/2) .  \nCorollary 1 . With high probability,  \nHG (Gn,1/2) ≤ (1 + o(1)) ~~ ~~√5 ! n ∼ 0.809n.  \nIt may be initially unclear whether the inequality in","cbCaikWG3wQyvAd5","https://ap.wps.com/l/cbCaikWG3wQyvAd5","pdf",285803,1,7,"English","en",105,"# Introduction\n## Proof Outline\n# Construction of the Auxiliary graph H","[{\"question\":\"What is the hat guessing number HG(G) of a graph?\",\"answer\":\"HG(G) is the largest k such that players placed on vertices can, using a predetermined simultaneous strategy with no communication, guarantee at least one correct guess of their own hat color for any coloring.\"},{\"question\":\"What parameters does the main upper bound depend on?\",\"answer\":\"The paper’s general bound depends on the number of vertices n (the order of the graph) and the maximum degree Δ.\"},{\"question\":\"What conclusions are drawn for the random graph Gn,1/2 and for high-degree graphs?\",\"answer\":\"For Gn,1/2, the hat guessing number is at most asymptotically c n with c around 0.809. More generally, graphs with maximum degree (1−ε)n cannot have HG(G) equal to (1−o(1))n.\"}]",1784194643,18,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"an-upper-bound-on-the-hat-guessing-number-of-graphs","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/an-upper-bound-on-the-hat-guessing-number-of-graphs/84295/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the hat guessing number HG(G) of a graph?","Question",{"text":75,"@type":76},"HG(G) is the largest k such that players placed on vertices can, using a predetermined simultaneous strategy with no communication, guarantee at least one correct guess of their own hat color for any coloring.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What parameters does the main upper bound depend on?",{"text":80,"@type":76},"The paper’s general bound depends on the number of vertices n (the order of the graph) and the maximum degree Δ.",{"name":82,"@type":73,"acceptedAnswer":83},"What conclusions are drawn for the random graph Gn,1/2 and for high-degree graphs?",{"text":84,"@type":76},"For Gn,1/2, the hat guessing number is at most asymptotically c n with c around 0.809. 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