On (d,1)-total numbers of graphs |
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Authors: | Ko-Wei Lih Daphne Der-Fen Liu |
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Institution: | a Institute of Mathematics, Academia Sinica, Nankang, Taipei 11529, Taiwan b Department of Mathematics, California State University, Los Angeles, Los Angeles, CA 90032, USA c Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China |
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Abstract: | A (d,1)-total labelling of a graph G assigns integers to the vertices and edges of G such that adjacent vertices receive distinct labels, adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least d. The span of a (d,1)-total labelling is the maximum difference between two labels. The (d,1)-total number, denoted , is defined to be the least span among all (d,1)-total labellings of G. We prove new upper bounds for , compute some for complete bipartite graphs Km,n, and completely determine all for d=1,2,3. We also propose a conjecture on an upper bound for in terms of the chromatic number and the chromatic index of G. |
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Keywords: | Channel assignment _method=retrieve& _eid=1-s2 0-S0012365X08006018& _mathId=si17 gif& _pii=S0012365X08006018& _issn=0012365X& _acct=C000053510& _version=1& _userid=1524097& md5=106c19a03e5bdbc2d54e0defabfeffb3')" style="cursor:pointer L(2" target="_blank">" alt="Click to view the MathML source" title="Click to view the MathML source">L(2 1)-labelling _method=retrieve& _eid=1-s2 0-S0012365X08006018& _mathId=si18 gif& _pii=S0012365X08006018& _issn=0012365X& _acct=C000053510& _version=1& _userid=1524097& md5=c14af28e17b05c4003847ae1752a8c92')" style="cursor:pointer (d" target="_blank">" alt="Click to view the MathML source" title="Click to view the MathML source">(d 1)-total labelling Chromatic number Chromatic index |
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