{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:48Z","timestamp":1753893828874,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We consider a random system of equations $x_i+x_j=b_{(i,j)} ({\\rm mod }2)$, $(x_u\\in \\{0,1\\},\\, b_{(u,v)}=b_{(v,u)}\\in\\{0,1\\})$, with the pairs $(i,j)$ from $E$, a symmetric subset of $[n]\\times [n]$.  $E$ is chosen uniformly at random among all such subsets of a given cardinality $m$; alternatively $(i,j)\\in E$ with a given probability $p$, independently of all other pairs. Also, given $E$, ${\\rm Pr}\\{b_{e}=0\\}={\\rm Pr}\\{b_e=1\\}$ for each $e\\in E$, independently of all other $b_{e\\prime}$.  It is well known that, as $m$ passes through $n\/2$ ($p$ passes through $1\/n$, resp.), the underlying random graph $G(n,\\#{\\rm edges}=m)$, ($G(n,{\\rm Pr}({\\rm edge})=p)$, resp.) undergoes a rapid transition, from essentially a forest of many small trees to a graph with one large, multicyclic, component in a sea of small tree components.  We should expect then that the solvability probability decreases precipitously in the vicinity of $m\\sim n\/2$ ($p\\sim 1\/n$), and indeed this probability is of order $(1-2m\/n)^{1\/4}$, for $m &lt; n\/2$ ($(1-pn)^{1\/4}$, for $p &lt; 1\/n$, resp.). We show that in a near-critical phase $m=(n\/2)(1+\\lambda n^{-1\/3})$ ($p=(1+\\lambda n^{-1\/3})\/n$, resp.), $\\lambda=o(n^{1\/12})$, the system is solvable with probability asymptotic to $c(\\lambda)n^{-1\/12}$, for some explicit function $c(\\lambda)&gt;0$. Mike Molloy noticed that the Boolean system with $b_e\\equiv 1$ is solvable iff the underlying graph is $2$-colorable, and asked whether this connection might be used to determine an order of probability of $2$-colorability in the near-critical case.  We answer Molloy's question affirmatively and show that, for $\\lambda=o(n^{1\/12})$, the probability of $2$-colorability is ${}\\lesssim  2^{-1\/4}e^{1\/8}c(\\lambda)n^{-1\/12}$, and asymptotic to $2^{-1\/4}e^{1\/8}c(\\lambda)n^{-1\/12}$ at a critical phase $\\lambda=O(1)$, and for $\\lambda\\to -\\infty$.<\/jats:p>","DOI":"10.37236\/364","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:11:46Z","timestamp":1578715906000},"source":"Crossref","is-referenced-by-count":5,"title":["How Frequently is a System of $2$-Linear Boolean Equations Solvable?"],"prefix":"10.37236","volume":"17","author":[{"given":"Boris","family":"Pittel","sequence":"first","affiliation":[]},{"given":"Ji-A","family":"Yeum","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,6,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r92\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r92\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:50:38Z","timestamp":1579305038000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r92"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,6,29]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/364","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,6,29]]},"article-number":"R92"}}