Pattern Periodic Coloring of Distance Graphs

Xuding Zhu
<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
Suppose D is a subset of Z. The distance graph G(Z, D) with distance set D is the graph with vertex set Z and two vertices x, y are adjacent if |x& y| # D. We introduce a coloring method for distance graphs, the pattern periodic coloring, and we shall compare this method with other general coloring methods of distance graphs. 1998 Academic Press INTRODUCTION Let Z be the set of all integers. For a subset D of Z, the distance graph G(Z, D) is the graph with vertex set Z in which two vertices x,
more &raquo; ... are adjacent if and only if |x& y| # D. The set D is called the distance set. Distance graphs are investigated in [1 8, 11, 12, 15 20]. It was motivated by the plane coloring problem: What is the least number of colors which can be used to color all points of the euclidean plane so that vertices of unit distance are colored with distinct colors. This problem is equivalent to determine the chromatic number of the distance graph on the plane R 2 with distance set D=[1]. It is known that the chromatic number of this distance graph is between 4 and 7 [10, 13], but there is no substantial progress on this problem in the last three decades. Distance graphs are also related to the channel assignment problem (or the T-coloring problem) and problems in number theory. We refer the readers to [2, 12, 20] for discussion of the relations among these problems. The problem of determining the chromatic numbers of distance graphs on the integers has received much recent attention [1, 2, 4, 5, 11 18, 20]. The cases that |D| =1 or 2 are easy [1, 17] . The case that |D| =3 is very complicated and has been completely settled only very recently [20] . The case that D is a subset of primes were discussed in [5, 15, 16] , and the problem is still open, although the case that D consists of four primes is solved by Voigt and Walther [16] . The case that D is an interval is solved
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