The hat guessing game on cactus graphs and cycles

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2024-10-03 DOI:10.1016/j.disc.2024.114272

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引用次数: 0

Abstract

We study the hat guessing game on graphs. In this game, a player is placed on each vertex v of a graph G and assigned a colored hat from

h (v)

possible colors. Each player makes a deterministic guess on their hat color based on the colors assigned to the players on neighboring vertices, and the players win if at least one player correctly guesses his assigned color. If there exists a strategy that ensures at least one player guesses correctly for every possible assignment of colors, the game defined by

〈 G, h 〉

is called winning. The hat guessing number of G is the largest integer q so that if

h (v) = q

for all

v \in G

then

〈 G, h 〉

is winning.

In this note, we determine whether

〈 G, h 〉

is winning for any h whenever G is a cycle, resolving a conjecture of Kokhas and Latyshev in the affirmative and extending it. We then use this result to determine the hat guessing number of every cactus graph, graphs in which every pair of cycles share at most one vertex.

查看原文本刊更多论文

仙人掌图形和周期的帽子猜谜游戏

我们研究的是图上的帽子猜谜游戏。在这个游戏中，一名玩家被安排在图 G 的每个顶点 v 上，并从 h(v) 种可能的颜色中分配一顶彩色帽子。每个玩家根据邻近顶点上玩家被分配的颜色，确定性地猜测自己帽子的颜色，如果至少有一个玩家猜对了自己被分配的颜色，则玩家获胜。如果存在一种策略能确保至少有一名玩家在每一种可能的颜色分配中都能猜对，那么〈G,h〉所定义的博弈就称为胜局。G 的猜帽数是最大整数 q，如果所有 v∈G 的 h(v)=q 则〈G,h〉获胜。在本说明中，我们将确定只要 G 是一个循环，〈G,h〉是否对任意 h 都获胜，从而解决科哈斯（Kokhas）和拉特谢夫（Latyshev）的一个猜想，并对其进行扩展。然后，我们利用这一结果确定了每个仙人掌图的帽子猜测数，在仙人掌图中，每对循环最多共享一个顶点。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.