Sigma Chromatic Number of the Middle Graph of Some Families of Graphs

One of the research topics involving vertex coloring of graphs that has become of inter- est to mathematicians in recent years is that of the sigma coloring. Introduced by Chartrand, Okamoto and Zhang in 2010, it is a type of vertex coloring that is neighbor-distinguishing; that is, it induces a...

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Main Author: Manamtam, Jay-R
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Published: Archīum Ateneo 2021
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Online Access:https://archium.ateneo.edu/theses-dissertations/533
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spelling ph-ateneo-arc.theses-dissertations-16592021-10-06T05:00:04Z Sigma Chromatic Number of the Middle Graph of Some Families of Graphs Manamtam, Jay-R One of the research topics involving vertex coloring of graphs that has become of inter- est to mathematicians in recent years is that of the sigma coloring. Introduced by Chartrand, Okamoto and Zhang in 2010, it is a type of vertex coloring that is neighbor-distinguishing; that is, it induces a coloring on the vertices of a graph. Let G be a nontrivial connected graph and let c : V (G) → N be a vertex coloring of G, where adjacent vertices may have the same color. For a vertex v of G, the color sum σ(v) of v is the sum of the colors of the vertices adjacent to v. The coloring c is said to be a sigma coloring of G if σ(u) 6= σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). On the other hand, the middle graph of a graph was introduced by T. Hamada and I. Yoshimura in 1976. In this study, we will study sigma coloring in relation to a unary graph operation called middle graph. We will show that the sigma chromatic number of the middle graph of paths, cycles, sunlet graphs, tadpole graphs, ladder graphs, and trian- gular snake graphs is 2. We also establish a lower bound for the sigma chromatic number of graphs containing an induced subgraph isomorphic to a complete graph appended with pendant vertices. Then, with this lower bound, we determine the sigma chromatic number of the middle graph of some general families of trees of height 2 such as star graphs, bistar graphs, and full m-ary trees. 2021-01-01T08:00:00Z text https://archium.ateneo.edu/theses-dissertations/533 Theses and Dissertations (All) Archīum Ateneo vertex coloring, sigma coloring, sigma chromatic number
institution Ateneo De Manila University
building Ateneo De Manila University Library
continent Asia
country Philippines
Philippines
content_provider Ateneo De Manila University Library
collection archium.Ateneo Institutional Repository
topic vertex coloring, sigma coloring, sigma chromatic number
spellingShingle vertex coloring, sigma coloring, sigma chromatic number
Manamtam, Jay-R
Sigma Chromatic Number of the Middle Graph of Some Families of Graphs
description One of the research topics involving vertex coloring of graphs that has become of inter- est to mathematicians in recent years is that of the sigma coloring. Introduced by Chartrand, Okamoto and Zhang in 2010, it is a type of vertex coloring that is neighbor-distinguishing; that is, it induces a coloring on the vertices of a graph. Let G be a nontrivial connected graph and let c : V (G) → N be a vertex coloring of G, where adjacent vertices may have the same color. For a vertex v of G, the color sum σ(v) of v is the sum of the colors of the vertices adjacent to v. The coloring c is said to be a sigma coloring of G if σ(u) 6= σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). On the other hand, the middle graph of a graph was introduced by T. Hamada and I. Yoshimura in 1976. In this study, we will study sigma coloring in relation to a unary graph operation called middle graph. We will show that the sigma chromatic number of the middle graph of paths, cycles, sunlet graphs, tadpole graphs, ladder graphs, and trian- gular snake graphs is 2. We also establish a lower bound for the sigma chromatic number of graphs containing an induced subgraph isomorphic to a complete graph appended with pendant vertices. Then, with this lower bound, we determine the sigma chromatic number of the middle graph of some general families of trees of height 2 such as star graphs, bistar graphs, and full m-ary trees.
format text
author Manamtam, Jay-R
author_facet Manamtam, Jay-R
author_sort Manamtam, Jay-R
title Sigma Chromatic Number of the Middle Graph of Some Families of Graphs
title_short Sigma Chromatic Number of the Middle Graph of Some Families of Graphs
title_full Sigma Chromatic Number of the Middle Graph of Some Families of Graphs
title_fullStr Sigma Chromatic Number of the Middle Graph of Some Families of Graphs
title_full_unstemmed Sigma Chromatic Number of the Middle Graph of Some Families of Graphs
title_sort sigma chromatic number of the middle graph of some families of graphs
publisher Archīum Ateneo
publishDate 2021
url https://archium.ateneo.edu/theses-dissertations/533
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