-labelings of subdivisions of graphs
Introduction
In the channel assignment problem, we need to assign frequency bands to transmitters, if two transmitters are too close, interference will occur if they attempt to transmit on close frequencies. In order to avoid this situation, the separation of the channels assigned to them must sufficient. Moreover, if two transmitters are close but not too close, the channel assigned must be different. This problem was known under the -labeling problem of a graph , where an -labeling of is a nonnegative integer assignment to the vertices of such that for all in , implies , and implies . A --labeling is an -labeling such that no label is greater than . The -labeling number of , denoted by (, and ), is the smallest number such that has a --labeling. This problem was first proposed by Griggs and Yeh [10] and has been studied extensively over the past years. Griggs and Yeh [10] showed that the -labeling problem is NP-complete for general graphs and proved that , where is the maximum degree of a vertex in . Chang and Kuo [7] gave a polynomial-time algorithm for the -labeling problem on trees. Gon çalves [9] proved that for any graph with . There are also many other results around this topic, for an overview on the subject, we refer the reader to the surveys in [5] and [23].
Whittlesty, Georges and Mauro [21] studied the -labeling of the incidence graph of , which is the graph obtained from by replacing each edge by a path of length . The -labeling of the incidence graph of is equivalent to an assignment of integers to each element of of such that: (i) any two adjacent vertices of receive different integers; (ii) any two adjacent edges of receive different integers; and (iii) a vertex and its incident edge receive integers that differ by at least in absolute value. This labeling is called -total labeling of , which was introduced by Havet and Yu in 2002 [11], [12] and generalized to the -total labeling of a graph . In [12], Havet and Yu obtained the bound and conjectured ( is the -total labeling number of ). It is sufficient to prove the conjecture for . Havet and Yu [12] completed the proof of the conjecture for by proving that if . Note that if , then the -total labeling is the traditional total coloring. The total coloring and -total labeling has been intensively studied in [1], [2], [3], [4], [8], [13], [17], [18], [19], [20], [22].
Given a graph and a function from to , the -subdivision of , denoted by , is the graph obtained from by replacing each edge in with a path (or simply, , if the function need not to be specified), where . If is a constant function; that is, for all , where is a constant, we use to replace . And we use to replace when for all for simplicity.
Lű [15] studied the -labeling problem of subdivisions of graphs. This problem can be viewed as a generalization of the -total labeling problem. Note that and for any graph . In [15], Lű proved that for all , for any graph . He also showed that for any graph . Based on these, he conjectured that for any graph . Lű and Lin [16] studied the -labeling problem of subdivisions of graphs and showed that for any graph with . From this, they deduced that for any graph . Karst et al. [14] showed that for any graph with . From this, they deduced that the conjecture proposed by Lű [15] is true, and, as a bonus, they obtained that for any graph with even . Chang et al. [6] studied the -labeling numbers of subdivisions of graphs, they proved that for any graph with . and that if and is a function from to so that for all , or if and is a function from to so that for all .
We study the -labeling numbers of subdivisions of graphs in this paper. We extend the results given in [6] and prove that when and . Based on this, we deduce that when and , where is a function from to so that for all .
Section snippets
Preliminaries
Throughout this paper, we assume that . We give some basic properties for the -labeling number of graphs in this section. Given a graph , a vertex in is called major if , where is the degree of in . The following two lemmas are easy to verify.
Lemma 1 If is a graph of maximum degree , then . Moreover, if and is a --labeling of , then or for every major vertex in .
Lemma 2 If is a --labeling
-labeling number of
We study the -labeling number of in this section. We first give some definitions and fix some notations that will be used later. Given a graph , a - walk of length in is a sequence of vertices and edges of the form , where , , for . Since the sequence of vertices of a walk determines the walk, we often use to denote a walk. A - trail is a walk in which all edges are distinct. For a vertex in , the neighborhood of
-labeling number of
We now consider the -labeling number of . From now on, for convenience, for two integers , and a set of integers , we use the symbol to denote the set , .
Given a graph together with an edge in , we can construct a graph with , . If is a function from to with , then for each with , the function defined on , which is defined by
Concluding notes
In this paper, we showed that if , then , when and is the function from to so that for all , or when and is a function from to so that for all . We believe that a more general result, which covers both Theorem 5, Theorem 7, is true. In fact, we have the following conjecture.
Conjecture 8 Let be a graph with maximum degree , and with , then whenever , and is a function from to so that
Acknowledgments
The authors would like to express their gratitude to the referees for their valuable comments and suggestions on improving the presentation of this paper.
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Cited by (1)
Exact square coloring of graphs resulting from some graph operations and products
2022, AKCE International Journal of Graphs and Combinatorics
- 1
Supported in part by the Ministry of Science and Technology of Taiwan under Grant No. MOST107-2115-M-003-005-MY2.
- 2
Supported in part by the Ministry of Science and Technology of Taiwan under Grant No. MOST106-2115-M-156-002.
- 3
Supported in part by the Ministry of Science and Technology of Taiwan under Grant No. MOST104-2115-M-259-002-MY2.
- 4
Supported in part by the Ministry of Science and Technology of Taiwan under Grant No. MOST106-2115-M-008-012-MY2.