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## Discussiones Mathematicae Graph Theory

2008 | 28 | 1 | 35-57
Tytuł artykułu

### Recognizable colorings of graphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let G be a connected graph and let c:V(G) → {1,2,...,k} be a coloring of the vertices of G for some positive integer k (where adjacent vertices may be colored the same). The color code of a vertex v of G (with respect to c) is the ordered (k+1)-tuple code(v) = (a₀,a₁,...,aₖ) where a₀ is the color assigned to v and for 1 ≤ i ≤ k, $a_i$ is the number of vertices adjacent to v that are colored i. The coloring c is called recognizable if distinct vertices have distinct color codes and the recognition number rn(G) of G is the minimum positive integer k for which G has a recognizable k-coloring. Recognition numbers of complete multipartite graphs are determined and characterizations of connected graphs of order n having recognition numbers n or n-1 are established. It is shown that for each pair k,n of integers with 2 ≤ k ≤ n, there exists a connected graph of order n having recognition number k. Recognition numbers of cycles, paths, and trees are investigated.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
35-57
Opis fizyczny
Daty
wydano
2008
otrzymano
2006-06-12
zaakceptowano
2007-10-30
Twórcy
autor
• Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
autor
• Department of Mathematics, and Computer Science, Drew University, Madison, NJ 07940, USA
autor
• Department of Mathematics, Grand Valley State University, Allendale, MI 49401, USA
autor
• Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
Bibliografia
• [1] L. Addario-Berry, R.E.L. Aldred, K. Dalal and B.A. Reed, Vertex colouring edge partitions, J. Combin. Theory (B) 94 (2005) 237-244, doi: 10.1016/j.jctb.2005.01.001.
• [2] M. Aigner and E. Triesch, Irregular assignments and two problems á la Ringel, in: Topics in Combinatorics and Graph Theory, R. Bodendiek and R. Henn, eds. (Physica, Heidelberg, 1990) 29-36.
• [3] M. Aigner, E. Triesch and Z. Tuza, Irregular assignments and vertex-distinguishing edge-colorings of graphs, Combinatorics' 90 (Elsevier Science Pub., New York, 1992) 1-9.
• [4] P.N. Balister, E. Gyori, J. Lehel and R.H. Schelp, Adjacent vertex distinguishing edge-colorings, SIAM J. Discrete Math. 21 (2007) 237-250, doi: 10.1137/S0895480102414107.
• [5] A.C. Burris, On graphs with irregular coloring number 2, Congr. Numer. 100 (1994) 129-140
• [6] A.C. Burris, The irregular coloring number of a tree, Discrete Math. 141 (1995) 279-283, doi: 10.1016/0012-365X(93)E0225-S.
• [7] G. Chartrand, H. Escuadro, F. Okamoto and P. Zhang, Detectable colorings of graphs, Util. Math. 69 (2006) 13-32.
• [8] G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congress. Numer. 64 (1988) 197-210.
• [9] G. Chartrand and L. Lesniak, Graphs & Digraphs: Fourth Edition (Chapman & Hall/CRC, Boca Raton, FL, 2005).
• [10] H. Escuadro, F. Okamoto and P. Zhang, A three-color problem in graph theory, Bull. Inst. Combin. Appl., to appear.
• [11] F. Harary and M. Plantholt, The point-distinguishing chromatic index, in: Graphs and Applications (Wiley, New York, 1985) 147-162.
• [12] M. Karoński, T. Łuczak and A. Thomason, Edge weights and vertex colours, J. Combin. Theory (B) 91 (2004) 151-157, doi: 10.1016/j.jctb.2003.12.001.
Typ dokumentu
Bibliografia
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