{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T06:00:50Z","timestamp":1768456850213,"version":"3.49.0"},"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>A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \\in V(G) \\cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\\phi_f(v)=\\sum_{e \\in E(v)}f(e)+f(v)$. A total weighting is proper if $\\phi_f(u) \\ne \\phi_f(v)$ for any edge $uv$ of $G$. A $(k,k')$-list assignment is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights, and assigns to each edge $e$ a set $L(e)$ of $k'$ permissible weights. We say $G$ is $(k,k')$-choosable if for any $(k,k')$-list assignment $L$, there is a proper total weighting $f$ of $G$ with $f(z) \\in L(z)$ for each $z \\in V(G) \\cup E(G)$. It was conjectured in [T. Wong and X. Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph is $(2,2)$-choosable and every graph with no isolated edge is $(1,3)$-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz. \u00a0This approach leads to conjectures on the permanent indices of matrices \u00a0$A_G$ and $B_G$ associated to a graph $G$. In this \u00a0paper, we establish a method that reduces the study of permanent of \u00a0matrices associated to a graph $G$ to the study of permanent of \u00a0matrices associated to induced subgraphs of $G$. Using this \u00a0reduction method, we show that if $G$ is a subcubic graph, or a $2$-tree, or a Halin graph, or a grid, then $A_G$ has permanent index $1$. As a consequence, these graphs are $(2,2)$-choosable.<\/jats:p>","DOI":"10.37236\/5494","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:23:02Z","timestamp":1578684182000},"source":"Crossref","is-referenced-by-count":3,"title":["Permanent Index of Matrices Associated with Graphs"],"prefix":"10.37236","volume":"24","author":[{"given":"Tsai-Lien","family":"Wong","sequence":"first","affiliation":[]},{"given":"Xuding","family":"Zhu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,2,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p25\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p25\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:05:25Z","timestamp":1579237525000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p25"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"https:\/\/doi.org\/10.37236\/5494","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,2,17]]},"article-number":"P1.25"}}