On the Sigma Value and Sigma Range of the Join of a Finite Number of Paths or Cycles

PhD Dissertation Defense
On the sigma value and sigma range of the join of a finite number of paths or cycles

by Marie Cris A. Bulay-og
PhD Mathematics Candidate

Date: Friday, 22 April 2022
Time: 3 pm
Venue: Online

Advisers:
Dr. Agnes Garciano & Dr. Reginaldo Marcelo
Ateneo de Manila University

Panelists:
Dr. Francis Joseph Campena
De La Salle University
Dr. Arnold Eniego
National University
Dr. Felix Muga II
Ateneo de Manila University
Dr. Mark Anthony Tolentino
Ateneo de Manila University

Let G be a simple connected graph and let c : V (G) → N be a vertex coloring of G. For a vertex v of G, the color sum of v, denoted by σ(v), is defined to be the sum of colors of the neighbors of v. If σ(u) ̸= σ(v) for any two adjacent vertices u and v in G, then c is called a sigma coloring of G. The minimum number of colors required in a sigma coloring of G is called the sigma chromatic number of G and it is denoted by σ(G). Suppose that σ(G) is k. The sigma value of G, denoted by ν(G), is the smallest positive integer s for which there exists a sigma coloring of G using k colors from the set {1,2,··· ,s}. On the other hand, the sigma range of G, denoted by ρ(G), is the smallest positive integer r for which there exists a sigma coloring of G using colors from the set {1,2,··· , r }. In this study, we determine the sigma value and sigma range of the join of a finite number of paths or cycles. For l ≥ 3, let G1 = Pl i=1 Pni , where 13 ≤ n1 ≤ n2 ≤ · · · ≤ nl , G2 = Pl i=1 Cni , where ni is even for all i and 4 ≤ n1 ≤ n2 ≤ ··· ≤ nl , and G3 = Pl i=1 Cni , where 11 ≤ n1 ≤ n2 ≤ ··· ≤ nl and at least one ni is odd. If ni+2−ni ≥ 4 for each 1 ≤ i ≤ l −2 in G1, then we show that ρ(G1) = ν(G1) = 2. On the other hand, if (n1,n2) ̸∈ {(4,4),(6,6)} in G2 and ni+2 −ni ≥ 4 for each 1 ≤ i ≤ l −2, then we show that ρ(G2) = ν(G2) = 2. Lastly, if ni+2 −ni ≥ 3 for each 1 ≤ i ≤ l − 2 in G3, then we show that ρ(G3) = ν(G3) = 3. If k is a positive integer, we also present necessary and sufficient conditions for each of the following to hold: for ρ(kPn) = ν(kPn) = 2, for ρ(kCn) = ν(kCn) = 2 when n is even, and for ρ(kCn) = ν(kCn) = 3 when n is odd.

Key Words: Sigma Coloring, Sigma Chromatic Number, Sigma Value, Sigma Range