Candy-passing Games on General Graphs, I

CAND Y-P ASSING GAMES ON GENERAL GRAPHS, I P AUL MYER KOMINERS AND SCOTT DUKE KOMINERS W e let G be an undir e c ted graph and denote the vertex and edge sets o f G by V ( G ) and E ( G ), re sp ectively . The c andy-p assing game on G is defined by the following rules : • At the beginning of the game, c > 0 candies are distributed among | V ( G ) | student s, each of who m is seated at some distinct vertex v ∈ V ( G ). • A whistle is sounded at a regular interv al. • Ea ch time the whistle is sounded, every student who is able to do so passes one candy to each of his neighbo rs. (If at the b eginning of this step a student ho lds fewer ca ndies than he has neighbors, he do es nothing.) The ca ndy-passing g a me was first introduced by T anton [3], who defined the game for cy clic G . T anton and W agon prov ed that if G is a n n -cycle then any candy-passing game on G with fewer than n candies terminates (se e [2]). T he first author [1] also studied the end be havior of candy-passing games on such G , showing that if the num ber o f ca ndies c is at leas t 3 n − 2, then the configur ation of candies even tually s tabilizes. Here, we undertake the firs t s tudy of the candy-pas s ing g a me on a r bitrary co n- nected graphs G . W e o btain a g eneral stabilization result which encompasses the first a utho r ’s [1] results for c ≥ 3 n . Preliminaries . W e call the interv al betw een soundings of the whis tle a r ound of candy-passing . Dropping the studen t metaphor, we will treat the candy piles as belo nging to the vertices of the g raph G . If, after some round, the amo unt of candy held by a given vertex will rema in constant throughout all future r ounds of the candy-passing ga me, that vertex is said to hav e s t abilize d . W e deno te the degre e of vertex v ∈ V ( G ) by deg( v ). Clearly , if some vertex v ∈ V ( G ) has k ≥ deg ( v ) candies at the b eginning of a round, that v ertex cannot end the round with more than k candies. Indeed, suc h a v ertex will pass deg( v ) pieces of candy to its neighbor s and can, at most, receive one piece o f c andy from each of its deg( v ) neigh b ors. Finally , we say that a v ertex v ∈ V ( G ) is abundant if it holds a t lea st 2 deg( v ) pieces o f candy . T his de finitio n implies: Lemma 1. After a finite numb er of r ounds o f the c andy-p assing ga me o n G , the set of abundant vertic es of G is fixe d and e ach abundant vertex has stabilize d. Pr o of. The total amount of candy o n a bundant vertices is nonincreasing . F urther- more, whenever an abundan t v ertex loses candy , that total decreases. Since the total amount of candy on abundant v ertices cannot fall be low zero, the amo unt 2000 Mathematics Subje ct Classific ation. 05C35 (Primary); 37B15 (Secondary). Key wor ds and phr ases. candy-passing, chip-firing, stabili zation, graph game. The s econd author gratefully ac knowledges th e support of a Harv ard Mathematics Depa rtment Highbridge F ell o wship. 1 2 P AUL MYER KOMINERS AND SCOTT DUKE KOMINERS of ca ndy that can b e lost by abundant v ertices m ust b e finite, so that the set o f abundant v ertices and the amount o f candy on each such vertex m ust even tually bec ome fixed.  Main Result. W e may now prove our stabiliza tion theor em: Theorem 2. L et G b e c onne cte d. In any c andy-p assing game on G with c ≥ 4 | E ( G ) | − | V ( G ) | c andies, every vertex v ∈ V ( G ) wil l eventual ly stabilize. Pr o of. As a consequence of Lemma 1, w e ma y assume that all c a ndy that will b e lost b y abundant vertices over the cour se of the game has b een lost, as this m ust happ en within finitely many rounds. If there are no abundant vertices at this po int , then the condition c ≥ 4 | E ( G ) | − | V ( G ) | implies c = 4 | E ( G ) | − | V ( G ) | and that every vertex v ∈ V ( G ) has 2 deg( v ) − 1 candies. In this case, all the v ertices of G hav e s tabilized. W e no w assume that at least one abunda nt vertex rema ins. (Unless we are in the situation a ddressed in the prior par agra ph, this is guara nteed by the conditio n c ≥ 4 | E ( G ) | − | V ( G ) | = P v ∈ V ( G ) (2 deg( v ) − 1).) This vertex ha s stabilized, and so it must be receiv ing c andy from all of its neighbor s ev ery r ound. Each of its neighbors v , therefore, m ust hold at le a st deg( v ) pieces of ca ndy every round. It follows that these neighbo rs m ust event ually stabilize, since these vertices pass candy every round and no such v ertex may end a r ound with mor e candy than it b egan with. By a n identical ar gument, the neighbors of these vertices must also pass ca ndy every round, and so they , to o , m ust even tually stabilize. As G is connected, con tinuing this ar gument shows that a ll the vertices o f G m ust even tually stabilize.  Remarks. When G is an n -cycle, the condition c ≥ 4 | E ( G ) | − | V ( G ) | is equiv a lent to the condition c ≥ 3 n . Our Theorem 2 generalize s the results of [1] for n -cycles with at least 3 n ca ndies. Mo re genera lly , if G is co nnected and k - re gu lar then the condition o f Theo rem 2 simplifies to c ≥ (2 k − 1) | V ( G ) | . References [1] P . M. Kominers, The candy-passing game for c ≥ 3 n − 2, Pi Mu Epsilon Journal 12 (8), 2008, pp. 459–460. [2] J. T anton, Solutions, The St. Mark’s Institute of Mathematics Newsletter , Nov em ber 2006. [3] J. T anton, T o day’s puzzler, The St. Mark’s Institute of Mathematics Newsletter , Nov ember 2006. Student, Depar tment of Ma thema tics, Massachusetts Institute of Technology c/o 8520 Burning Tree Road Bethesda, MD 2 0817 E-mail addr ess : pkoms@mit.ed u Student, Depar tment of Ma thema tics, Har v ard University E-mail addr ess : kominers@fas .harvard.edu