{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T03:11:14Z","timestamp":1787368274483,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>Motivated by $L(p,q)$-labelings of graphs, we introduce a notion of $\\lambda$-graphs: a $\\lambda$-graph G is a graph with two types of edges: 1-edges and x-edges. For a parameter $x\\in[0,1]$, a proper labeling of G is a labeling of vertices of G by nonnegative reals such that the labels of the endvertices of a 1-edge differ by at least 1 and the labels of the endvertices of an x-edge differ by at least x; $\\lambda_G(x)$ is the smallest real such that G has a proper labeling by labels from the interval $[0,\\lambda_G(x)]$. We study properties of the function $\\lambda_G(x)$ for finite and infinite $\\lambda$-graphs and establish the following results: if the function $\\lambda_G(x)$ is well defined, then it is a piecewise linear function of x with finitely many linear parts. Surprisingly, the set $\\Lambda(\\alpha,\\beta)$ of all functions $\\lambda_G$ with $\\lambda_G(0)=\\alpha$ and $\\lambda_G(1)=\\beta$ is finite for any $\\alpha\\le\\beta$. We also prove a tight upper bound on the number of segments for finite $\\lambda$-graphs G with convex functions $\\lambda_G(x)$.<\/jats:p>","DOI":"10.1137\/040619545","type":"journal-article","created":{"date-parts":[[2007,9,12]],"date-time":"2007-09-12T18:02:15Z","timestamp":1189620135000},"page":"688-706","source":"Crossref","is-referenced-by-count":5,"title":["Labelings of Graphs with Fixed and Variable Edge-Weights"],"prefix":"10.1137","volume":"21","author":[{"given":"Robert","family":"Babilon","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"V\u00edt","family":"Jel\u00ednek","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Kr\u00e1l'","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pavel","family":"Valtr","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,9,12]]},"reference":[{"key":"R1","first-page":"41","volume":"144","author":"Agnarsson G.","year":"2000","journal-title":"Proceedings of the Thirty-First South Eastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 2000) Congr. Numer."},{"key":"R2","unstructured":"T. M. Apostol,\n                      Modular Functions and Dirichlet Series in Number Theory\n                      , Springer-Verlag, New York, 1997."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"H. L. Bodlaender, T. Kloks, R. B. Tan, and J. van Leeuwen,\n                      $\\lambda$-coloring of graphs\n                      , in Proc. STACS'00, G. Goos, J. Hartmanis, and J. van Leeuwen, eds., Lecture Notes in Comput. Sci. 1770, Springer-Verlag, Berlin, 2000, pp. 395\u2013406.","DOI":"10.1007\/3-540-46541-3_33"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(99)00400-8"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480193245339"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"J. H. Conway and R. K. Guy,\n                      The Book of Numbers\n                      , Springer-Verlag, Berlin, 1996.","DOI":"10.1007\/978-1-4612-4072-3"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(00)00387-5"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"D. A. Fotakis, S. E. Nikoletseas, V. G. Papadopoulou, and P. G. Spirakis,\n                      NP-completeness results and efficient approximations for radiocoloring in planar graphs\n                      , in Mathematical Foundations of Computer Science 2000 (Bratislava), Lecture Notes in Comput. Sci. 1893, B. Rovan, ed., Springer-Verlag, Berlin, 2000, pp. 363\u2013372.","DOI":"10.1007\/3-540-44612-5_32"},{"key":"R9","first-page":"81","author":"Gon\u00e7alves D.","year":"2005","journal-title":"Discrete Math. Theoret. Comput. Sci."},{"key":"R10","unstructured":"R. L. Graham, D. E. Knuth, and O. Patashnik,\n                      Concrete Mathematics: A Foundation for Computer Science\n                      , 2nd ed., Addison-Wesley, Reading, MA, 1994."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480105446708"},{"key":"R12","unstructured":"J. R. Griggs and T. X. Jin,\n                      Real Number Graph Labellings of Paths and Cycles\n                      , in preparation."},{"key":"R13","unstructured":"J. R. Griggs and T. X. Jin,\n                      Real Number Graph Labellings of Infinite Graphs\n                      , in preparation."},{"key":"R14","unstructured":"J. R. Griggs and D. Kr\u00e1l',\n                      Graph Labellings with Variable Weights:\n                      -1ptA Survey\n                      , Discrete Appl. Math., to appear."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/0405048"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1109\/PROC.1980.11899"},{"key":"R17","unstructured":"G. H. Hardy and E. M. Wright,\n                      An Introduction to the Theory of Numbers\n                      , Clarendon Press, New York, 1979."},{"key":"R18","unstructured":"T. X. Jin, Ph.D. thesis, University of South Carolina, Columbia, SC, in preparation."},{"key":"R19","unstructured":"J.H. Kang,\n                      $L(2,1)$-labeling of\n                      3\n                      -regular Hamiltonian graphs\n                      , submitted."},{"key":"R20","unstructured":"J.H. Kang,\n                      $L(2,1)$-Labelling of\n                      3\n                      -Regular Hamiltonian Graphs\n                      , Ph.D. thesis, University of Illinois, Urbana-Champaign, IL, 2004."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2004.01.020"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480103424079"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/040619636"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480101399449"},{"key":"R25","doi-asserted-by":"crossref","first-page":"Resear","DOI":"10.37236\/1769","volume":"11","author":"Leese R. A.","year":"2004","journal-title":"Electron. J. Combin.","ISSN":"https:\/\/id.crossref.org\/issn\/1077-8926","issn-type":"print"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"C. McDiarmid,\n                      Discrete mathematics and radio channel assignment\n                      , in Recent Advances in Algorithms and Combinatorics CMS Books Math., C. Linhares-Salas and B. Reed, eds., Ouvrages Math. SMC 11, Springer, New York, 2003, pp. 27\u201363.","DOI":"10.1007\/0-387-22444-0_2"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(02)00821-X"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480191223178"},{"key":"R29","unstructured":"I. Vardi,\n                      Computational Recreations in Mathematica\n                      , Addison-Wesley, Redwood City, CA, 1991."},{"key":"R30","unstructured":"E. W. Weisstein,\n                      Farey Sequence\n                      , http:\/\/mathworld.wolfram.com\/FareySequence.html."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/040619545","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:54:45Z","timestamp":1787320485000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/040619545"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/040619545"],"URL":"https:\/\/doi.org\/10.1137\/040619545","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}