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Distance Constrained Labelings of K4minor Free Graphs
, 2006
"... Motivated by previous results on distance constrained labelings and coloring of squares of K4minor free graphs, we show that for every p> = q> = 1, there exists \Delta 0 such that every K4minor free graph G with maximum degree \Delta> = \Delta 0 has an L(p, q)labeling of span at most qb3 ..."
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Motivated by previous results on distance constrained labelings and coloring of squares of K4minor free graphs, we show that for every p> = q> = 1, there exists \Delta 0 such that every K4minor free graph G with maximum degree \Delta> = \Delta 0 has an L(p, q)labeling of span at most qb3\Delta (G)/2c. The obtained bound is the best possible.
On Colorings of graph fractional powers
, 2009
"... For any k ∈ N, the k−subdivision of graph G is a simple graph G 1 k, which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n. In this regard, we investigate chromatic nu ..."
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For any k ∈ N, the k−subdivision of graph G is a simple graph G 1 k, which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n. In this regard, we investigate chromatic number and clique number of fractional power of graphs. Also, we conjecture that χ(G m n) = ω(G m n) provided that G is a connected graph with < 1. It is also shown that this conjecture is true in some special cases. ∆(G) ≥ 3 and m
Generalized Powers of Graphs and Their Algorithmic Use
, 2006
"... Motivated by the frequency assignment problem in heterogeneous multihop radio networks, where different radio stations may have different transmission ranges, we introduce two new types of coloring of graphs, which generalize the wellknown DistancekColoring. Let G =(V,E) be a graph modeling a r ..."
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Motivated by the frequency assignment problem in heterogeneous multihop radio networks, where different radio stations may have different transmission ranges, we introduce two new types of coloring of graphs, which generalize the wellknown DistancekColoring. Let G =(V,E) be a graph modeling a radio network, and assume that each vertex v of G has its own transmission radius r(v), a nonnegative integer. We define rcoloring (r +coloring) ofG as an assignment Φ: V ↦ → {0, 1, 2,...} of colors to vertices such that Φ(u) =Φ(v) implies dG(u, v)>r(v)+r(u)(dG(u, v)>r(v)+r(u) + 1, respectively). The rColoring problem (the r +Coloring problem) asks for a given graph G and a radiusfunction r: V ↦ → N ∪{0}, to find an rcoloring (an r +coloring, respectively) of G with minimum number of colors. Using a new notion of generalized powers of graphs, we investigate the complexity of the rColoring and r +Coloring problems on several families of graphs.
On injective colourings of chordal graphs
 Lecture Notes in Computer Sci
"... Abstract. We show that one can compute the injective chromatic number of a chordal graph G at least as efficiently as one can compute the chromatic number of (G−B)2, where B are the bridges of G. In particular, it follows that for strongly chordal graphs and socalled power chordal graphs the inje ..."
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Abstract. We show that one can compute the injective chromatic number of a chordal graph G at least as efficiently as one can compute the chromatic number of (G−B)2, where B are the bridges of G. In particular, it follows that for strongly chordal graphs and socalled power chordal graphs the injective chromatic number can be determined in polynomial time. Moreover, for chordal graphs in general, we show that the decision problem with a fixed number of colours is solvable in polynomial time. On the other hand, we show that computing the injective chromatic number of a chordal graph is NPhard; and unless NP = ZPP, it is hard to approximate within a factor of n1/3−ǫ, for any ǫ> 0. For split graphs, this is best possible, since we show that the injective chromatic number of a split graph is 3 napproximable. (In the process, we correct a result of Agnarsson et al. on inapproximability of the chromatic number of the square of a split graph.) 1
DISTANCE TWO LABELING FOR MULTISTOREY GRAPHS
"... An L (2, 1)labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k}such that f(x)f(y)  ≥2 if d(x, y) =1 and  f(x) f(y)  ≥1 if d(x, y) =2. The L (2, 1)labeling number λ (G) or span of G is the smallest k such t ..."
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An L (2, 1)labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k}such that f(x)f(y)  ≥2 if d(x, y) =1 and  f(x) f(y)  ≥1 if d(x, y) =2. The L (2, 1)labeling number λ (G) or span of G is the smallest k such that there is a f with max {f (v) : vє V(G)} = k. In this paper we introduce a new type of graph called multistorey graph. The distance two labeling of multistorey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multistorey of simple graph are discussed. AMS Subject Classification: 05C78
New results in ttone coloring of graphs
, 2012
"... A ttone kcoloring of G assigns to each vertex of G a set of t colors from {1,..., k} so that vertices at distance d share fewer than d common colors. The ttone chromatic number of G, denoted τt(G), is the minimum k such that G has a ttone kcoloring. Bickle and Phillips showed that always τ2(G) ..."
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A ttone kcoloring of G assigns to each vertex of G a set of t colors from {1,..., k} so that vertices at distance d share fewer than d common colors. The ttone chromatic number of G, denoted τt(G), is the minimum k such that G has a ttone kcoloring. Bickle and Phillips showed that always τ2(G) ≤ [∆(G)]2 + ∆(G), but conjectured that in fact τ2(G) ≤ 2∆(G) + 2; we confirm this conjecture when ∆(G) ≤ 3 and also show that always τ2(G) ≤
Distance constrained labelings of planar graphs with no short cycles
, 2007
"... Motivated by a conjecture of Wang and Lih, we show that every planar graph of girth at least 7 and maximum degree ∆ ≥ 190 + 2⌈p/q ⌉ has an L(p, q)labeling of span at most 2p+q∆−2. Since the optimal span of an L(p, 1)labeling of an infinite ∆regular tree is 2p+∆ − 2, the obtained bound is the bes ..."
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Motivated by a conjecture of Wang and Lih, we show that every planar graph of girth at least 7 and maximum degree ∆ ≥ 190 + 2⌈p/q ⌉ has an L(p, q)labeling of span at most 2p+q∆−2. Since the optimal span of an L(p, 1)labeling of an infinite ∆regular tree is 2p+∆ − 2, the obtained bound is the best possible for any p ≥ 1 and q = 1.