scholarly journals Odd Wheels Are Not Odd-Distance Graphs

2021 ◽  
Author(s):  
Gábor Damásdi
Keyword(s):  
Distance Graphs ◽  
Wheel Graph ◽  
Odd Cycle

AbstractAn odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle. We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane so that the lengths of the edges are odd integers.

New Results on Edge Rotation Distance Graphs

2016 ◽  
Vol 6 (1) ◽  
pp. 81
Author(s):  
Medha Itagi Huilgol ◽  
Chitra Ramprakash
Keyword(s):  
Distance Graphs

S-packing colorings of distance graphs G(Z,{2,t})

2021 ◽  
Vol 298 ◽  
pp. 143-154
Author(s):  
Boštjan Brešar ◽  
Jasmina Ferme ◽  
Karolína Kamenická
Keyword(s):  
Distance Graphs

Density of sets with missing differences and applications

2021 ◽  
Vol 71 (3) ◽  
pp. 595-614
Author(s):  
Ram Krishna Pandey ◽  
Neha Rai
Keyword(s):  
Number Theory ◽  
Asymptotic Density ◽  
Theory Problem ◽  
Distance Graphs ◽  
Positive Integers ◽  
Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(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 find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

Independence numbers and chromatic numbers of some distance graphs

2015 ◽  
Vol 51 (2) ◽  
pp. 165-176 ◽  
Author(s):  
A. V. Bobu ◽  
O. A. Kostina ◽  
A. E. Kupriyanov
Keyword(s):  
Chromatic Numbers ◽  
Distance Graphs ◽  
Independence Numbers

scholarly journals Chromatic numbers of exact distance graphs

2019 ◽  
Vol 134 ◽  
pp. 143-163 ◽  
Author(s):  
Jan van den Heuvel ◽  
H.A. Kierstead ◽  
Daniel A. Quiroz
Keyword(s):  
Chromatic Numbers ◽  
Distance Graphs

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

Bounds on Borsuk Numbers in Distance Graphs of a Special Type

2021 ◽  
Vol 57 (2) ◽  
pp. 136-142
Author(s):  
A. V. Berdnikov ◽  
A.M. Raigorodskii
Keyword(s):  
Distance Graphs
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