## Taiwanese Journal of Mathematics

### Maximal Density of Sets with Missing Differences and Various Coloring Parameters of Distance Graphs

#### Abstract

For a given set $M$ of positive integers, a well-known problem of Motzkin asked to determine the maximal asymptotic density of $M$-sets, denoted by $\mu(M)$, where an $M$-set is a set of non-negative integers in which no two elements differ by an element in $M$. In 1973, Cantor and Gordon found $\mu(M)$ for $|M| \leq 2$. Partial results are known in the case $|M| \geq 3$ including results in the case when $M$ is an infinite set. This number theory problem is also related to various types of coloring problems of the distance graphs generated by $M$. In particular, it is known that the reciprocal of the fractional chromatic number of the distance graph generated by $M$ is equal to the value $\mu(M)$ when $M$ is finite. Motivated by the families $M = \{ a,b,a+b \}$ and $M = \{ a,b,a+b,b-a \}$ discussed by Liu and Zhu, we study two families of sets $M$, namely, $M = \{ a,b,b-a,n(a+b) \}$ and $M = \{ a,b,a+b,n(b-a) \}$. For both of these families, we find some exact values and some bounds on $\mu(M)$. We also find bounds on the fractional and circular chromatic numbers of the distance graphs generated by these families. Furthermore, we determine the exact values of chromatic number of the distance graphs generated by these two families.

#### Article information

Source
Taiwanese J. Math., Advance publication (2020), 15 pages.

Dates
First available in Project Euclid: 22 April 2020

https://projecteuclid.org/euclid.twjm/1587542423

Digital Object Identifier
doi:10.11650/tjm/200403

Subjects
Primary: 11B05: Density, gaps, topology
Secondary: 05C15: Coloring of graphs and hypergraphs

#### Citation

Pandey, Ram Krishna; Rai, Neha. Maximal Density of Sets with Missing Differences and Various Coloring Parameters of Distance Graphs. Taiwanese J. Math., advance publication, 22 April 2020. doi:10.11650/tjm/200403. https://projecteuclid.org/euclid.twjm/1587542423

#### References

• J. Barajas and O. Serra, Distance graphs with maximum chromatic number, Discrete Math. 308 (2008), no. 8, 1355–1365.
• D. G. Cantor and B. Gordon, Sequences of integers with missing differences, J. Combinatorial Theory Ser. A 14 (1973), 281–287.
• G. J. Chang, D. D.-F. Liu and X. Zhu, Distance graphs and $T$-colorings, J. Combin. Theory Ser. B 75 (1999), no. 2, 259–269.
• J.-J. Chen, G. J. Chang and K.-C. Huang, Integral distance graphs, J. Graph Theory 25 (1997), no. 4, 287–294.
• D. Collister and D. D.-F. Liu, Study of $\kappa(D)$ for $D = \{ 2,3,x,y \}$, in: Combinatorial Algorithms, 250–261, Lecture Notes in Computer Science, Springer, Cham, 2015.
• T. W. Cusick, View-obstruction problems in $n$-dimensional geometry, J. Combinatorial Theory Ser. A 16 (1974), 1–11.
• J. R. Griggs and D. D.-F. Liu, The channel assignment problem for mutually adjacent sites, J. Combin. Theory Ser. A 68 (1994), no. 1, 169–183.
• S. Gupta, Sets of integers with missing differences, J. Combin. Theory Ser. A 89 (2000), no. 1, 55–69.
• S. Gupta and A. Tripathi, Density of $M$-sets in arithmetic progression, Acta Arith. 89 (1999), no. 3, 255–257.
• N. M. Haralambis, Sets of integers with missing differences, J. Combinatorial Theory Ser. A 23 (1977), no. 1, 22–33.
• A. Kemnitz and H. Kolberg, Coloring of integer distance graphs, Discrete Math. 191 (1998), no. 1-3, 113–123.
• A. Kemnitz and M. Marangio, Chromatic numbers of integer distance graphs, Discrete Math. 233 (2001), no. 1-3, 239–246.
• ––––, Colorings and list colorings of integer distance graphs, Congr. Numer. 151 (2001), 75–84.
• D. D.-F. Liu, From rainbow to the lonely runner: A survey on coloring parameters of distance graphs, Taiwanese J. Math. 12 (2008), no. 4, 851–871.
• D. D.-F. Liu and G. Robinson, Sequences of integers with three missing separations, European J. Combin. 85 (2020), 103056, 11 pp.
• D. D.-F. Liu and A. Sutedja, Chromatic number of distance graphs generated by the sets $\{ 2,3,x,y \}$, J. Comb. Optim. 25 (2013), no. 4, 680–693.
• D. D.-F. Liu and X. Zhu, Fractional chromatic number and circular chromatic number for distance graphs with large clique size, J. Graph Theory 47 (2004), no. 2, 129–146.
• ––––, Fractional chromatic number of distance graphs generated by two-interval sets, European J. Combin. 29 (2008), no. 7, 1733–1743.
• T. S. Motzkin, Unpublished problems collection.
• R. K. Pandey and A. Tripathi, A note on a problem of Motzkin regarding density of integral sets with missing differences, J. Integer Seq. 14 (2011), no. 6, Article 11.6.3, 8 pp.
• ––––, On the density of integral sets with missing differences from sets related to arithmetic progressions, J. Number Theory 131 (2011), no. 4, 634–647.
• ––––, A note on the density of $M$-sets in geometric sequence, Ars Combin. 119 (2015), 221–224.
• J. H. Rabinowitz and V. K. Proulx, An asymptotic approach to the channel assignment problem, SIAM J. Algebraic Discrete Methods 6 (1985), no. 3, 507–518.
• A. Srivastava, R. K. Pandey and O. Prakash, On the maximal density of integral sets whose differences avoiding the weighted Fibonacci numbers, Integers 17 (2017), A48, 19 pp.
• ––––, Motzkin's maximal density and related chromatic numbers, Unif. Distrib. Theory 13 (2018), no. 1, 27–45.
• M. Voigt, Colouring of distance graphs, Ars Combin. 52 (1999), 3–12.
• J. M. Wills, Zwei Sätze über inhomogene diophantische Approximation von Irrationalzahlen, Monatsh. Math. 71 (1967), 263–269.
• X. Zhu, Circular chromatic number: A survey, Discrete Math. 229 (2001), no. 1-3, 371–410.
• ––––, Circular chromatic number of distance graphs with distance sets of cardinality $3$, J. Graph Theory 41 (2002), no. 3, 195–207.