{"id":324350,"date":"2016-11-18T15:53:09","date_gmt":"2016-11-18T23:53:09","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&#038;p=324350"},"modified":"2018-10-16T20:19:54","modified_gmt":"2018-10-17T03:19:54","slug":"g-parking-functions-acyclic-orientations-spanning-trees","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/g-parking-functions-acyclic-orientations-spanning-trees\/","title":{"rendered":"G-Parking Functions, Acyclic Orientations and Spanning Trees"},"content":{"rendered":"<p>Given an undirected graph <span id=\"mmlsi38\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si38.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=730bcbce3e087f853e5c7e3397a06f89\">G=(V,E)<\/span><\/span>, and a designated vertex <span id=\"mmlsi39\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si39.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=1ac04dcdefda8d759da37f28774dc761\">q\u2208V<\/span><\/span>, the notion of a <span id=\"mmlsi40\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si40.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=d445f19ce30dd9ca3554588cb2f8f0e0\">G<\/span><\/span>-parking function (with respect to <span id=\"mmlsi41\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si41.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=c74d0558252bfc0fcfc7a71791265a37\">q<\/span><\/span>) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study the properties of <em>maximum \u00a0<\/em><span id=\"mmlsi42\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si42.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=68d0dde0e41c73591eb42ba38f1d10cc\">G<\/span><\/span>-parking functions and provide a new bijection between them and the set of spanning trees of <span id=\"mmlsi43\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si43.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=04c21cf57e15568591d04be7131c5149\">G<\/span><\/span> with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the discrete <span id=\"mmlsi44\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si44.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=8de60f5ee6d6118168873676610f8afa\">n<\/span><\/span>-cube <span id=\"mmlsi45\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si45.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=f58ad30b181c943bc9d7477c9f0c50b5\">Q<sub>n<\/sub><\/span><\/span>. We present the article in an expository self-contained form, since we found the combinatorial aspects of <span id=\"mmlsi46\" class=\"mathmlsrc\"><span class=\"formulatext stixSupport mathImg\" title=\"Click to view the MathML source\" data-mathurl=\"\/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X10000142&_mathId=si46.gif&_user=111111111&_pii=S0012365X10000142&_rdoc=1&_issn=0012365X&md5=f40b73205cdf13fe3bc48c0c47ef8d4e\">G<\/span><\/span>-parking functions somewhat scattered in the literature, typically treated in conjunction with sandpile models and closely related chip-firing games.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given an undirected graph G=(V,E), and a designated vertex q\u2208V, the notion of a G-parking function (with respect to q) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study the [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Elsevier Science Publishers B. V. 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