{"id":"https://openalex.org/W2767768384","doi":"https://doi.org/10.1155/2017/5897049","title":"Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers","display_name":"Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2767768384","doi":"https://doi.org/10.1155/2017/5897049","mag":"2767768384"},"language":"lv","primary_location":{"id":"doi:10.1155/2017/5897049","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2017/5897049","pdf_url":"http://downloads.hindawi.com/journals/ijmms/2017/5897049.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"http://downloads.hindawi.com/journals/ijmms/2017/5897049.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088218909","display_name":"Patcharapan Jumnongnit","orcid":null},"institutions":[{"id":"https://openalex.org/I179193067","display_name":"Khon Kaen University","ror":"https://ror.org/03cq4gr50","country_code":"TH","type":"education","lineage":["https://openalex.org/I179193067"]}],"countries":["TH"],"is_corresponding":false,"raw_author_name":"Patcharapan Jumnongnit","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand","institution_ids":["https://openalex.org/I179193067"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023616108","display_name":"Kittikorn Nakprasit","orcid":"https://orcid.org/0000-0002-0421-3631"},"institutions":[{"id":"https://openalex.org/I179193067","display_name":"Khon Kaen University","ror":"https://ror.org/03cq4gr50","country_code":"TH","type":"education","lineage":["https://openalex.org/I179193067"]}],"countries":["TH"],"is_corresponding":true,"raw_author_name":"Kittikorn Nakprasit","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"],"raw_orcid":"https://orcid.org/0000-0002-0421-3631","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand","institution_ids":["https://openalex.org/I179193067"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5023616108"],"corresponding_institution_ids":["https://openalex.org/I179193067"],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.19628471,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"2017","issue":null,"first_page":"1","last_page":"4"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5435293912887573},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.33321985602378845},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.25821653008461}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5435293912887573},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.33321985602378845},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25821653008461}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1155/2017/5897049","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2017/5897049","pdf_url":"http://downloads.hindawi.com/journals/ijmms/2017/5897049.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:hin:jijmms:5897049","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/IJMMS/2017/5897049.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:d6e8d8b352a048d8bae34518e97fdfa5","is_oa":true,"landing_page_url":"https://doaj.org/article/d6e8d8b352a048d8bae34518e97fdfa5","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences, Vol 2017 (2017)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2017/5897049","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2017/5897049","pdf_url":"http://downloads.hindawi.com/journals/ijmms/2017/5897049.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320326818","display_name":"University of Phayao","ror":"https://ror.org/00a5mh069"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2767768384.pdf","grobid_xml":"https://content.openalex.org/works/W2767768384.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W599177077","https://openalex.org/W1773291652","https://openalex.org/W1967155492","https://openalex.org/W1967183009","https://openalex.org/W1976013119","https://openalex.org/W1983048542","https://openalex.org/W1998350558","https://openalex.org/W1999957357","https://openalex.org/W2057166029","https://openalex.org/W2069762242","https://openalex.org/W2071649283","https://openalex.org/W2108756413","https://openalex.org/W2296149237","https://openalex.org/W2925167782"],"related_works":["https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2053286651","https://openalex.org/W2181743346","https://openalex.org/W2187401768","https://openalex.org/W2181413294","https://openalex.org/W2989452537","https://openalex.org/W2052122378","https://openalex.org/W2544423928","https://openalex.org/W2062023542"],"abstract_inverted_index":{"Let":[0,25],"<mml:math":[1,8,21,26,39,49,53,61,67,81,86,94,101,121,141,144,148,155,165,176,180,191,196,204,208,213,221,227,235,243,252,257,267,276,284,289,302,305,311,337],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[2,9,22,27,40,50,54,62,68,82,87,95,102,122,142,145,149,156,166,177,181,192,197,205,209,214,222,228,236,244,253,258,263,268,277,285,290,298,303,306,312,338],"id=\"M1\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>":[3],"be":[4,19],"a":[5,20,37,72,90,152,201,247,261,293],"graph":[6],"and":[7,42,115,162,179,226,330],"id=\"M2\"><mml:mi>\u03d5</mml:mi><mml:mo>:</mml:mo><mml:mi>V</mml:mi><mml:mo":[10],"stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo":[11,13,64,110,130,224,238,279,320,346],"stretchy=\"false\">)</mml:mo><mml:mo>\u222a</mml:mo><mml:mi>E</mml:mi><mml:mo":[12],"stretchy=\"false\">)</mml:mo><mml:mo>\u2192</mml:mo><mml:mo":[14],"stretchy=\"false\">{</mml:mo><mml:mn":[15],"mathvariant=\"normal\">1,2</mml:mn><mml:mo>,</mml:mo><mml:mn":[16],"mathvariant=\"normal\">3</mml:mn><mml:mo>,</mml:mo><mml:mo>\u2026</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo":[17],"stretchy=\"false\">}</mml:mo></mml:math>":[18],"id=\"M3\"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-total":[23],"coloring.":[24,77],"id=\"M4\"><mml:mi>w</mml:mi><mml:mo":[28],"stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mo":[29,58,171,216,218,272],"stretchy=\"false\">)</mml:mo></mml:math>":[30,59,131,219,225,233,273],"denote":[31],"the":[32,118,134],"sum":[33,74,92,249,295],"of":[34,136,147,158,207],"color":[35],"on":[36,133],"vertex":[38,175],"id=\"M5\"><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:math>":[41],"colors":[43],"assigned":[44],"to":[45,48,160],"edges":[46,163],"incident":[47],"id=\"M6\"><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:math>.":[51],"If":[52],"id=\"M7\"><mml:mi>w</mml:mi><mml:mo":[55],"stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo":[56,270],"stretchy=\"false\">)</mml:mo><mml:mo>\u2260</mml:mo><mml:mi>w</mml:mi><mml:mo":[57,271],"whenever":[60,220,234],"id=\"M8\"><mml:mi>u</mml:mi><mml:mi>v</mml:mi><mml:mo>\u2208</mml:mo><mml:mi>E</mml:mi><mml:mo":[63],"stretchy=\"false\">)</mml:mo></mml:math>,":[65],"then":[66],"id=\"M9\"><mml:mrow><mml:mi>\u03d5</mml:mi></mml:mrow></mml:math>":[69],"is":[70,98,151,200,308],"called":[71],"neighbor":[73,91,248,294],"distinguishing":[75,93,250,296],"total":[76,202],"The":[78,281],"smallest":[79,282],"integer":[80,283],"id=\"M10\"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>":[83],"such":[84,211,265,287],"that":[85,212,242,266,288],"id=\"M11\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>":[88],"has":[89,246,260,292],"id=\"M12\"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-total":[96],"coloring":[97,203],"denoted":[99,309],"by":[100,310,328,332],"id=\"M13\"><mml:mi":[103],"mathvariant=\"normal\">t</mml:mi><mml:mi":[104,124],"mathvariant=\"normal\">n</mml:mi><mml:mi":[105,125],"mathvariant=\"normal\">d</mml:mi><mml:msub><mml:mrow><mml:mi":[106,126],"mathvariant=\"normal\">i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo":[107,127],"stretchy=\"false\">\u2211</mml:mo><mml:mrow><mml:mo>":[108,128,316,342],"</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:msub><mml:mo":[109,129],"stretchy=\"false\">)</mml:mo></mml:math>.":[111,239,280,321,347],"In":[112,322],"2014,":[113],"Dong":[114,329],"Wang":[116,331],"obtained":[117],"results":[119,327,335],"about":[120],"id=\"M14\"><mml:mi":[123],"depending":[132],"value":[135],"maximum":[137],"average":[138],"degree.":[139],"A":[140,194],"id=\"M15\"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-assignment":[143],"id=\"M16\"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>":[146],"id=\"M17\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>":[150],"list":[153],"assignment":[154],"id=\"M18\"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>":[157],"integers":[159],"vertices":[161],"with":[164],"id=\"M19\"><mml:mfenced":[167],"open=\"|\"":[168,183],"close=\"|\"":[169,184],"separators=\"|\"><mml:mrow><mml:mi>L</mml:mi><mml:mo":[170,185],"stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:math>":[172,187],"for":[173,188,274,300,336],"each":[174,189],"id=\"M20\"><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:math>":[178],"id=\"M21\"><mml:mfenced":[182],"stretchy=\"false\">(</mml:mo><mml:mi>e</mml:mi><mml:mo":[186,230,232],"edge":[190],"id=\"M22\"><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:math>.":[193],"total-":[195,251],"id=\"M23\"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>":[198],"-coloring":[199,255],"id=\"M24\"><mml:mrow><mml:mi>\u03d5</mml:mi></mml:mrow></mml:math>":[206],"id=\"M25\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>":[210],"id=\"M26\"><mml:mi>\u03d5</mml:mi><mml:mo":[215],"stretchy=\"false\">)</mml:mo><mml:mo>\u2208</mml:mo><mml:mi>L</mml:mi><mml:mo":[217,231],"id=\"M27\"><mml:mi>v</mml:mi><mml:mo>\u2208</mml:mo><mml:mi>V</mml:mi><mml:mo":[223],"id=\"M28\"><mml:mi>\u03d5</mml:mi><mml:mo":[229],"id=\"M29\"><mml:mi>e</mml:mi><mml:mo>\u2208</mml:mo><mml:mi>E</mml:mi><mml:mo":[237],"We":[240],"state":[241],"id=\"M30\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>":[245],"id=\"M31\"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>":[254],"if":[256],"id=\"M32\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>":[259],"total-<mml:math":[262,297],"id=\"M33\"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>-coloring":[264],"id=\"M34\"><mml:mi>w</mml:mi><mml:mo":[269],"all":[275],"id=\"M35\"><mml:mi>u</mml:mi><mml:mi>v</mml:mi><mml:mo>\u2208</mml:mo><mml:mi>E</mml:mi><mml:mo":[278],"id=\"M36\"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>":[286],"id=\"M37\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>":[291],"id=\"M38\"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>-coloring":[299],"every":[301],"id=\"M39\"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-assignment":[304],"id=\"M40\"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>":[307],"id=\"M41\"><mml:mi":[313],"mathvariant=\"normal\">C</mml:mi><mml:msubsup><mml:mrow><mml:mi":[314,340],"mathvariant=\"normal\">h</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo":[315,341],"</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi":[317,343],"mathvariant=\"normal\">\u2032</mml:mi><mml:mi":[318,344],"mathvariant=\"normal\">\u2032</mml:mi></mml:mrow></mml:msubsup><mml:mo":[319,345],"this":[323],"paper,":[324],"we":[325],"strengthen":[326],"giving":[333],"analogous":[334],"id=\"M42\"><mml:mi":[339]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}