Geodesic
On Closed Modular Colorings of Trees
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 411-428.

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Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − u, v. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) →ℤ_k be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c^′ : V (G) →ℤ_k defined by c′(v) = Σ_u ∈ N[v] c(u) for each v ∈ V (G), where N[v] is the closed neighborhood of v. Then c is called a closed modular k-coloring if c^′(u) c′(v) in ℤ_k for all pairs u, v of adjacent vertices that are not true twins. The minimum k for which G has a closed modular k-coloring is the closed modular chromatic number mc(G) of G. The closed modular chromatic number is investigated for trees and determined for several classes of trees. For each tree T in these classes, it is shown that mc (T) = 2 or mc(T) = 3. A closed modular k-coloring c of a tree T is called nowhere-zero if c(x) 0 for each vertex x of T. It is shown that every tree of order 3 or more has a nowhere-zero closed modular 4-coloring.
Keywords: trees, closed modular k-coloring, closed modular chromatic number 
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Phinezy, Bryan; Zhang, Ping. On Closed Modular Colorings of Trees. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 411-428. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a13/

[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] G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congr. Numer. 64 (1988) 197-210.

[3] G. Chartrand, F. Okamoto and P. Zhang, The sigma chromatic number of a graph, Graphs Combin. 26 (2010) 755-773. doi:10.1007/s00373-010-0952-7

[4] G. Chartrand, B. Phinezy and P. Zhang, On closed modular colorings of regular graphs, Bull. Inst. Combin. Appl. 66 (2012) 7-32.

[5] G. Chartrand, L. Lesniak and P. Zhang, Graphs & Digraphs, 5th Edition (Chapman & Hall/CRC, Boca Raton, 2010).

[6] G. Chartrand and P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC, Boca Raton, 2008). doi:10.1201/9781584888017

[7] H. Escuadro, F. Okamoto and P. Zhang, Vertex-distinguishing colorings of graphs- A survey of recent developments, AKCE Int. J. Graphs Comb. 4 (2007) 277-299.

[8] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 16 (2009) #.DS6

[9] R.B. Gnana Jothi, Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University 1991.

[10] S.W. Golomb, How to number a graph, in: Graph Theory and Computing (Academic Press, New York, 1972) 23-37.

[11] R. Jones, K. Kolasinski and P. Zhang, A proof of the modular edge-graceful trees conjecture, J. Combin. Math. Combin. Comput., to appear.

[12] M. Karónski, T. Luczak and A. Thomason, Edge weights and vertex colours, J. Combin. Theory (B) 91 (2004) 151-157. doi:10.1016/j.jctb.2003.12.001

[13] A. Rosa, On certain valuations of the vertices of a graph, in: Theory of Graphs, Pro. Internat. Sympos. Rome 1966 (Gordon and Breach, New York, 1967) 349-355.