On Irregular Colorings of Unicyclic Graph Family CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 7(4) (2023), Pages 503-512 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: July 01, 2022 Reviewed: July 24, 2022 Accepted: August 15, 2022 DOI: http://dx.doi.org/10.18860/ca.v7i4.16917 On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana*, Dafik, Qurrotul A’yun, Robiatul Adawiyah, Ridho Alfarisi Mathematics Education Study Program, Faculty of Teacher Training and Education University of Jember, Indonesia Email: arika.fkip@unej.ac.id ABSTRACT An Irregular coloring is a proper coloring where each vertex on a graph must have a different code. The color code of a vertex v is π‘π‘œπ‘‘π‘’(𝑣) = (π‘Ž0,π‘Ž1,π‘Ž2, . . . ,π‘Žπ‘˜) where π‘Ž0 = 𝑐(𝑣) and π‘Žπ‘– = 𝑖,1 ≀ 𝑖 ≀ π‘˜ is the number of vertices that are adjacent to 𝑣 and colored 𝑖. The minimum k-color used in irregular coloring is called the irregular chromatic number and denoted by πœ’π‘–π‘Ÿ. The type of research used in this research is exploratory research. In this paper, we discuss the irregular chromatic number of the bull graph, pan graph, sun graph, peach graph, and caveman graph. Copyright Β© 2023 by Authors, Published by CAUCHY Group. This is an open access article under the CC BY- SA License (https://creativecommons.org/licenses/by-sa/4.0/) Keywords: irregular coloring; irregular chromatic number; unicyclic graph INTRODUCTION A graph 𝐺 = (𝑉,𝐸) is a pair of a vertex set denoted by 𝑉(𝐺) and an edge set (may be empty) denoted by 𝐸(𝐺). More detail definitions and some properties can be seen in [1]. In that year, a four-color theorem which discusses maps coloring was discovered. This theorem states that there is a separation of a plane into adjacent areas resulting in an image called a map. The maximum colors which is needed to color the area on the map so that two neighboring areas do not have the same color is four [2]. π‘π‘œπ‘‘π‘’(𝑣) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜) where π‘Ž0 = 𝑐(𝑣) and π‘Žπ‘– = 𝑖,(1 ≀ 𝑖 ≀ π‘˜) is the number of vertices that are adjacent to 𝑣, and colored 𝑣, where 𝑐(𝑣) is the color at vertex 𝑣. The minimum color used in irregular coloring is called the irregular chromatic number and denoted by πœ’π‘–π‘Ÿ [2]. An Irregular coloring is included proper coloring of G, it follows that [2] πœ’(𝐺) ≀ πœ’π‘–π‘Ÿ(𝐺). The following are the theorems, corollaries, and observations that will be used in this paper. Lemma 1. [2] For every pair π‘Ž,𝑏 of integers with 2 ≀ π‘Ž ≀ 𝑏, there is a connected graph 𝐺 with πœ’(𝐺) = π‘Ž and πœ’π‘–π‘Ÿ(𝐺) = 𝑏. Corollary 1. [2] For every graph 𝐺, πœ”(𝐺) ≀ πœ’(𝐺) The clique number πœ”(𝐺) of a graph 𝐺 is the maximum order of a complete subgraph of 𝐺. Observation 1.[2] Let 𝑐 be a (proper) vertex coloring of a nontrivial graph 𝐺 and let 𝑒 and 𝑣 be two distinct vertices of 𝐺. a. If 𝑐(𝑒) β‰  𝑐(𝑣), then π‘π‘œπ‘‘π‘’(𝑒) β‰  π‘π‘œπ‘‘π‘’ (𝑣) b. If 𝑑𝑒𝑔𝐺𝑒 β‰  𝑑𝑒𝑔𝐺𝑣, then π‘π‘œπ‘‘π‘’(𝑒) β‰  π‘π‘œπ‘‘π‘’ (𝑣) c. If 𝑐 is irregular coloring and 𝑁(𝑒) = 𝑁(𝑣), 𝑐(𝑒) β‰  𝑐(𝑣) http://dx.doi.org/10.18860/ca.v7i4.16917 mailto:arika.fkip@unej.ac.id https://creativecommons.org/licenses/by-sa/4.0/ On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana 504 Lemma 2. [3] Let 𝑐 be an irregular k-coloring of the vertices of a graph 𝐺. The number of different possible color codes of the vertices of degree r in G is π‘˜( π‘Ÿ + (π‘˜ + 1) βˆ’ 1 π‘Ÿ ) = π‘˜( π‘Ÿ + π‘˜ βˆ’ 2 π‘Ÿ ) Corollary 2. [3] If c is an irregular k-coloring of a nontrivial connected graph G, then G contains at most π‘˜( π‘Ÿ + π‘˜ βˆ’ 2 π‘Ÿ ) vertices of degree r. Corollary 3. [3] If πœ’π‘–π‘Ÿ(𝐺) = π‘˜ where 𝑛 β‰₯ 3, then 𝑛 ≀ (π‘˜)( π‘˜ 2 ) = π‘˜2(π‘˜βˆ’1) 2 Corollary 4.[3] Let π‘˜ β‰₯ 3 be an integer, πœ’π‘–π‘Ÿ(𝐢𝑛) β‰₯ π‘˜ for all integers n. Such that (π‘˜ βˆ’ 1)2(π‘˜ βˆ’2) +2 2 ≀ 𝑛 ≀ π‘˜2(π‘˜ βˆ’ 1) 2 Propotition 1.[3] πœ’π‘–π‘Ÿ(𝐢𝑛) = 4 if n is even and πœ’π‘–π‘Ÿ(𝐢𝑛) = 3 if n is odd for 3 ≀ 𝑛 ≀ 9. Lemma 3. [3] Let π‘˜ β‰₯ 3, if 𝑛 = π‘˜2(π‘˜βˆ’1) 2 , then πœ’π‘–π‘Ÿ(πΆπ‘›βˆ’1) β‰₯ π‘˜ + 1 A. Rohoni et. al have discussed irregular coloring in the double wheel graph family [4], fan graphs families [5], and wheel related graphs [6]. Avudainayaki et. al [7] has discussed irregular coloring on central and middle graphs of double star graphs, Anderson et. Al [8] discussed irregular coloring on regular graphs, and Kristiana et. al [9] has discussed local irregularity coloring of some family graphs. There are some previous results of irregular coloring of bipartite graph dan tree graph family [10], star families [11], some generalized graph [12], and Mycielskian Graphs [13]. In this paper, we observed the irregular coloring of bull graph, pan graph, sun graph, peach graph, and caveman graph. The bull Graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges [14]. The Pan Pan graph is a graph obtained by combining a cycle graph Cn with a singleton star graph K1[15]. The sun graph of order 2𝑛 is a cycle 𝐢𝑛 with an edge terminating in a pendent vertex attached to each vertex[16]. A peach graph is a circular graph Cm which has 𝑛 pendants at one vertex, which is at vertex π‘₯1 contained in the cycle graph[17]. A caveman graph arises by modifying a set of fully connected clusters (caves) by removing one edge from each cluster and using it to connect to a neighboring one such that the clusters form a single loop [18]. METHOD The type of research used in this research is exploratory research. This research is research conducted to dig up data and find new things that you want to know so that the results obtained can be used as knowledge development. The following is a description of the steps taken to determine irregular chromatic numbers: (1) determining the graph that will be researched; (2) determining the cardinality of the graph that is used as research; (3) do vertex coloring of the researched graph; (4) create a code from each vertex in the graph. Specifically, π‘π‘œπ‘‘π‘’ (𝑣) = (π‘Ž0,π‘Ž1,…,π‘Žπ‘˜) =, where π‘Ž0 = 𝑐(𝑣) and π‘Žπ‘– (1 ≀ 𝑖 ≀ π‘˜) are the number of vertices of color 𝑖 adjacent to 𝑣; (5) if the codes are the same for each vertex, proceed to step 3. (6) determining irregular chromatic numbers. RESULT AND DISCUSSION This research is focused on finding irregular chromatic numbers in the unicyclic graph family, including bull graph, pan graph, sun graph, peach graph, and caveman graph. On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana 505 Theorem 1. The irregular chromatic number of bull graph π‘©πŸ‘,𝒏 is 𝒏 + 𝟏. Proof: This graph has the vertex set 𝑉(𝐡3,𝑛) = {π‘₯𝑖| 1 ≀ 𝑖 ≀ 3} βˆͺ {𝑦2𝑖|1 ≀ 𝑖 ≀ 𝑛} βˆͺ {𝑦3𝑖|1 ≀ 𝑖 ≀ 𝑛} and edge set is 𝐸(𝐡3,𝑛) = {π‘₯1π‘₯𝑖| 2 ≀ 𝑖 ≀ 3} βˆͺ{π‘₯2π‘₯3} βˆͺ{π‘₯2𝑦2𝑖| 1 ≀ 𝑖 ≀ 𝑛} βˆͺ{π‘₯3𝑦3𝑖| 1 ≀ 𝑖 ≀ 𝑛}.|𝑉(𝐡3,𝑛)| = 2𝑛 + 3 and |𝐸(𝐡3,𝑛)| = 2𝑛 + 3. We need to prove the upper bound πœ’π‘–π‘Ÿ(𝐡3,𝑛) ≀ 𝑛 + 1 and the lower bound πœ’π‘–π‘Ÿ(𝐡3,𝑛) β‰₯ 𝑛 + 1. First, we prove that the lower bound of the irregular chromatic number of bull graph is πœ’π‘–π‘Ÿ(𝐡3,𝑛) β‰₯ 𝑛 + 1. Assume that πœ’π‘–π‘Ÿ(𝐡3,𝑛) < 𝑛 + 1. We have some condition as follows. (1) 𝑐(π‘₯𝑖) = 𝑐(𝑦2𝑖) and 𝑐(π‘₯𝑖) 𝑐(𝑦2𝑖) ∈ 𝐸(𝐡3,𝑛). It's contradicting. (2) 𝑐(π‘₯𝑖) = 𝑐(𝑦3𝑖) and 𝑐(π‘₯𝑖) 𝑐(𝑦3𝑖) ∈ 𝐸(𝐡3,𝑛). It's contradicting. (3) 𝑐(π‘₯2π‘˜) = 𝑐(𝑦2𝑙) and π‘˜ β‰  𝑙. It's contradicting. (4) 𝑐(π‘₯3π‘˜) = 𝑐(𝑦3𝑙) and π‘˜ β‰  𝑙. It's contradicting. Based on (1), (2), (3), and (4), we get the lower bound of the irregular coloring on a bull graph is πœ’π‘–π‘Ÿ(𝐡3,𝑛) β‰₯ 𝑛 + 1. Furthermore, we prove that the upper bound of the irregular chromatic number of the bull graph is πœ’π‘–π‘Ÿ(𝐡3,𝑛) ≀ 𝑛 + 1. The color function 𝑐 on this graph is defined as follows. 𝑐(π‘₯𝑖) = 𝑖,1 ≀ 𝑖 ≀ 3 𝑐(𝑦2𝑖) = { 𝑖 𝑖 = 1 𝑖 + 1 2 ≀ 𝑖 ≀ 𝑛 𝑐(𝑦3𝑖) = { 𝑖 1 ≀ 𝑖 ≀ 2 𝑖 + 1 3 ≀ 𝑖 ≀ 𝑛 Based on the color function, the code for each vertex in the bull graph is obtained as follows. π‘π‘œπ‘‘π‘’(π‘₯1) = (𝑐(π‘₯1),0,1,1,0,0,…,0⏟ 𝑛+1 ) π‘π‘œπ‘‘π‘’(π‘₯2) = (𝑐(π‘₯2),2,0,2,1,1,…,1⏟ 𝑛+1 ) π‘π‘œπ‘‘π‘’(π‘₯3) = (𝑐(π‘₯3),2,1,0,1,1,…,1⏟ 𝑛+1 ) π‘π‘œπ‘‘π‘’(𝑦2𝑖) = (𝑐(𝑦2𝑖),0,1,0,0,0,…,0⏟ 𝑛+1 ) π‘π‘œπ‘‘π‘’(𝑦3𝑖) = (𝑐(𝑦3𝑖),0,0,1,0,0,…,0⏟ 𝑛+1 ) Based on the color function and vertex code obtained, each neighboring vertex has a different color and each vertex in the graph has a different color code. Therefore, the upper bound of the irregular chromatic number on the bull graph is πœ’π‘–π‘Ÿ(𝐡3,𝑛) ≀ 𝑛 + 1. The lower and upper bound of the irregular chromatic number of bull graph is 𝑛 + 1 ≀ πœ’π‘–π‘Ÿ(𝐡3,𝑛) ≀ 𝑛 + 1. So, πœ’π‘–π‘Ÿ(𝐡3,𝑛) = 𝑛 + 1. Theorem 2. Let (π’Œβˆ’πŸ)( π’Œ βˆ’ 𝟏 𝟐 ) + 𝟏 ≀ 𝒏 ≀ π’Œ( π’Œ 𝟐 ) and π’Œ β‰₯ πŸ’. Irregular chromatic number of pan graph 𝑷𝒂𝒏 is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = { 3 π‘“π‘œπ‘Ÿ 𝑛 π‘œπ‘‘π‘‘,3 ≀ 𝑛 ≀ 9 3 π‘“π‘œπ‘Ÿ 𝑛 = 4 4 π‘“π‘œπ‘Ÿ 𝑛 𝑒𝑣𝑒𝑛,6 ≀ 𝑛 ≀ 9 On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana 506 Proof: This graph has the vertex set 𝑉(π‘ƒπ‘Žπ‘›) = {π‘₯𝑖|1 ≀ 𝑖 ≀ 𝑛} βˆͺ{𝑦} and the edge set is 𝐸(π‘ƒπ‘Žπ‘›) = {π‘₯𝑖π‘₯𝑖+1|1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1} βˆͺ{π‘₯1π‘₯𝑛} βˆͺ {π‘₯𝑖𝑦}. |𝑉(π‘ƒπ‘Žπ‘›)| = 𝑛 + 1 and |𝐸(π‘ƒπ‘Žπ‘›)| = 𝑛 + 1. The pan graph contains the 𝐢𝑛 subgraph where 𝐢𝑛 is a cycle graph. In this case, the 𝐢𝑛 subgraph is labeled according to the 𝐢𝑛 graph. There are three cases in this proof. Case 1. πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = 3,π‘“π‘œπ‘Ÿ 𝑛 π‘œπ‘‘π‘‘,3 ≀ 𝑛 ≀ 9 We need to prove the upper bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3 and the lower bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 3. First, we prove that the lower bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 3. Assume that πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) < 3, for example πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = 2, we get the color on π‘₯𝑖 is 1,2,1,2, . . . ,1 or 2,1,2, . . . ,2 periodically. Since 𝑐(π‘₯1) = 𝑐(π‘₯𝑛) and π‘₯1π‘₯𝑛 ∈ 𝐸(π‘ƒπ‘Žπ‘›). This contradicts the definition of proper coloring where each pair of adjacent vertices must have different colors. Therefore, πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 3. Furthermore, we prove that the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3 where 3 ≀ 𝑛 ≀ 9. Define the color function 𝑐:𝑣 β†’ {1,2,3,…,πœ’π‘–π‘Ÿ},𝑆 = {1,2,3,…,πœ’π‘–π‘Ÿ} and 𝑝,π‘ž ∈ 𝑁. π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1 ∈ 𝑉(π‘ƒπ‘Žπ‘›), with 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1), 𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) ∈ 𝑆. If 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) β‰  𝑐(π‘₯(π‘ž(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(π‘ž+2(π‘šπ‘œπ‘‘ 𝑛))+1), then 𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1) = 𝑐(π‘₯(π‘ž+1(π‘šπ‘œπ‘‘ 𝑛))+1). The color of y is 𝑐(𝑦), for 𝑐(𝑦) ∈ 𝑆, 𝑐(π‘₯𝑖) β‰  𝑐(𝑦), π‘₯𝑖𝑦 ∈ 𝐸(π‘ƒπ‘Žπ‘›). Based on the color function, it can be written, if it is known that π‘π‘œπ‘‘π‘’(π‘₯𝑝) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), and π‘π‘œπ‘‘π‘’(π‘₯π‘ž) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), π‘π‘œπ‘‘π‘’(π‘₯𝑝) β‰  π‘π‘œπ‘‘π‘’(π‘₯π‘ž). We get the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3. The lower and upper bound of the irregular chromatic number of pan graph is 3 ≀ πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3. So, πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = 3. Case 2. πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = 3,𝑛 = 4 We need to prove the upper bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3 and the lower bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 3. First, we prove that the lower bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 3. Assume that πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) < 3, for example πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = 2. Define the color function 𝑐:𝑣 β†’ {1,2} is 𝑐(π‘₯1) = 1,𝑐(π‘₯2) = 2,𝑐(π‘₯3) = 1,𝑐(π‘₯4) = 2,𝑐(𝑦) = 2. The color code obtained is π‘π‘œπ‘‘π‘’(π‘₯1) = (1,0,3), π‘π‘œπ‘‘π‘’(π‘₯2) = (2,2,0),π‘π‘œπ‘‘π‘’(π‘₯3) = (1,0,2),π‘π‘œπ‘‘π‘’(π‘₯4) = (2,2,0),π‘π‘œπ‘‘π‘’(𝑦) = (2,1,0). Based on the code obtained, there is the same code, namely π‘π‘œπ‘‘π‘’(π‘₯2) = π‘π‘œπ‘‘π‘’(π‘₯4). This contradicts the definition of irregular colorin that each vertex in a graph must have a different color code. So, πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 3. Furthermore, we prove that the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3. Define the color function 𝑐:𝑣 β†’ {1,2,3} is 𝑐(π‘₯1) = 1,𝑐(π‘₯2) = 3,𝑐(π‘₯3) = 1,𝑐(π‘₯4) = 2,𝑐(𝑦) = 2. The color code obtained is π‘π‘œπ‘‘π‘’(π‘₯1) = (1,0,2,1), π‘π‘œπ‘‘π‘’(π‘₯2) = (3,2,0,0),π‘π‘œπ‘‘π‘’(π‘₯3) = (1,0,1,1),π‘π‘œπ‘‘π‘’(π‘₯4) = (2,2,0,0),π‘π‘œπ‘‘π‘’(𝑦) = (2,1,0,0). So, it can be concluded that πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3. We get the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3. The lower and upper bound of the irregular chromatic number of pan graph is 3 ≀ πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 3. So, πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = 3. Case 3. πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = 4,π‘“π‘œπ‘Ÿ 𝑛 𝑒𝑣𝑒𝑛,6 ≀ 𝑛 ≀ 9 We need to prove upper bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 4 and lower bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 4. First, we prove that the lower bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 4. According to Lemma 3, if π‘˜ = 3, then πœ’π‘–π‘Ÿ(πΆπ‘›βˆ’1) β‰₯ 4. We get an odd n, so On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana 507 𝑛 βˆ’ 1 is even. Since Pan has the subgraph Cn, where Cn is a cycle graph labeled as Cn. So, based on Lemma 3, we can conclude that πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 4. Next, πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ 4. Furthermore, we prove that the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 4 where 3 ≀ 𝑛 ≀ 9. Define the color function 𝑐:𝑣 β†’ {1,2,3,…,πœ’π‘–π‘Ÿ},𝑆 = {1,2,3,…,πœ’π‘–π‘Ÿ} and 𝑝,π‘ž ∈ 𝑁. π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1 ∈ 𝑉(π‘ƒπ‘Žπ‘›), with 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) ∈ 𝑆. If 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) β‰  𝑐(π‘₯(π‘ž(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(π‘ž+2(π‘šπ‘œπ‘‘ 𝑛))+1), then 𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1) = 𝑐(π‘₯(π‘ž+1(π‘šπ‘œπ‘‘ 𝑛))+1). The color on y is 𝑐(𝑦), for 𝑐(𝑦) ∈ 𝑆, 𝑐(π‘₯𝑖) β‰  𝑐(𝑦), π‘₯𝑖𝑦 ∈ 𝐸(π‘ƒπ‘Žπ‘›). Based on the color function, it can be written, if it is known that π‘π‘œπ‘‘π‘’(π‘₯𝑝) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), and π‘π‘œπ‘‘π‘’(π‘₯π‘ž) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), π‘π‘œπ‘‘π‘’(π‘₯𝑝) β‰  π‘π‘œπ‘‘π‘’(π‘₯π‘ž). So, we get the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 4. The lower and upper bound of the irregular chromatic number of pan graph is 4 ≀ πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ 4. So, πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = 4. Figure 1. πœ’π‘–π‘Ÿ(π‘ƒπ‘Ž7) = 3 Theorem 3. Let (π’Œβˆ’πŸ)( π’Œ βˆ’ 𝟏 𝟐 ) + 𝟏 ≀ 𝒏 ≀ π’Œ( π’Œ 𝟐 ) Irregular chromatic number of pan graph 𝑷𝒂𝒏 for 𝒏 β‰₯ πŸ‘,π’Œ β‰₯ πŸ’ is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = { π‘˜ π‘“π‘œπ‘Ÿ 𝑛 β‰  π‘˜( π‘˜ 2 ) βˆ’1 π‘˜ + 1 π‘“π‘œπ‘Ÿ 𝑛 = π‘˜( π‘˜ 2 ) βˆ’1 Proof: This graph has the vertex set 𝑉(π‘ƒπ‘Žπ‘›) = {π‘₯𝑖|1 ≀ 𝑖 ≀ 𝑛} βˆͺ{𝑦} and the edge set is 𝐸(π‘ƒπ‘Žπ‘›) = {π‘₯𝑖π‘₯𝑖+1|1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1} βˆͺ{π‘₯1π‘₯𝑛} βˆͺ {π‘₯𝑖𝑦}. |𝑉(π‘ƒπ‘Žπ‘›)| = 𝑛 + 1 and |𝐸(π‘ƒπ‘Žπ‘›)| = 𝑛 + 1. The pan graph contains the 𝐢𝑛 subgraph where 𝐢𝑛 is a cycle graph. In this case, the 𝐢𝑛 subgraph is labeled according to the 𝐢𝑛 graph. There are two cases in this proof. Case 1. πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = π‘˜,π‘“π‘œπ‘Ÿ 𝑛 β‰  π‘˜( π‘˜ 2 ) βˆ’ 1 We need to prove upper bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ π‘˜ and lower bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ π‘˜. First, we prove that the lower bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ π‘˜. According to Corollary 4, if π‘˜ β‰₯ 3 is an integer, then πœ’π‘–π‘Ÿ(𝐢𝑛) β‰₯ π‘˜. Since π‘ƒπ‘Žπ‘› has the subgraph 𝐢𝑛, where 𝐢𝑛 is a cycle graph labeled as 𝐢𝑛. Therefore, we get πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ π‘˜. Next, we prove that the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ π‘˜. Color function 𝑐:𝑣 β†’ {1,2,3,…,πœ’π‘–π‘Ÿ},𝑆 = {1,2,3,…,πœ’π‘–π‘Ÿ} and 𝑝,π‘ž ∈ 𝑁. π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1 ∈ 𝑉(π‘ƒπ‘Žπ‘›), with 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) ∈ 𝑆. On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana 508 If 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) β‰  𝑐(π‘₯(π‘ž(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(π‘ž+2(π‘šπ‘œπ‘‘ 𝑛))+1), then 𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1) = 𝑐(π‘₯(π‘ž+1(π‘šπ‘œπ‘‘ 𝑛))+1). The color on y is 𝑐(𝑦), for 𝑐(𝑦) ∈ 𝑆, 𝑐(π‘₯𝑖) β‰  𝑐(𝑦), π‘₯𝑖𝑦 ∈ 𝐸(π‘ƒπ‘Žπ‘›). Based on the color function, it can be written, if it is known that π‘π‘œπ‘‘π‘’(π‘₯𝑝) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), and π‘π‘œπ‘‘π‘’(π‘₯π‘ž) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), π‘π‘œπ‘‘π‘’(π‘₯𝑝) β‰  π‘π‘œπ‘‘π‘’(π‘₯π‘ž). So, we get the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ π‘˜. The lower and upper bound of the irregular chromatic number of pan graph is π‘˜ ≀ πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ π‘˜. So, πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = π‘˜. Case 2. πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = π‘˜ + 1,π‘“π‘œπ‘Ÿ 𝑛 = π‘˜( π‘˜ 2 ) βˆ’ 1 We need to prove upper bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ π‘˜ + 1 and lower bound πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ π‘˜ + 1. First, we prove that the lower bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ π‘˜ +1. According to Lemma 3, if 𝑛 = π‘˜2(π‘˜+1) 2 = π‘˜( π‘˜ 2 ), then πœ’π‘–π‘Ÿ(πΆπ‘›βˆ’1) β‰₯ π‘˜ + 1. Since π‘ƒπ‘Žπ‘› has the subgraph 𝐢𝑛, where 𝐢𝑛 is a cycle graph labeled as 𝐢𝑛. Therefore, if 𝑛 βˆ’ 1, then π‘˜( π‘˜ 2 ) βˆ’ 1. So, if 𝑛 = π‘˜( π‘˜ 2 ) βˆ’1, then πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) β‰₯ π‘˜ + 1. Next, we prove that the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ π‘˜ +1. Color function 𝑐:𝑣 β†’ {1,2,3,…,πœ’π‘–π‘Ÿ},𝑆 = {1,2,3,…,πœ’π‘–π‘Ÿ} and 𝑝,π‘ž ∈ 𝑁. π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1 ∈ 𝑉(π‘ƒπ‘Žπ‘›),with 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1), 𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) ∈ 𝑆. If 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) β‰  𝑐(π‘₯(π‘ž(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(π‘ž+2(π‘šπ‘œπ‘‘ 𝑛))+1), then 𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1) = 𝑐(π‘₯(π‘ž+1(π‘šπ‘œπ‘‘ 𝑛))+1). The color on y is 𝑐(𝑦), for 𝑐(𝑦) ∈ 𝑆, 𝑐(π‘₯𝑖) β‰  𝑐(𝑦), π‘₯𝑖𝑦 ∈ 𝐸(π‘ƒπ‘Žπ‘›). Based on the color function, it can be written, if it is known that π‘π‘œπ‘‘π‘’(π‘₯𝑝) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), and π‘π‘œπ‘‘π‘’(π‘₯π‘ž) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), π‘π‘œπ‘‘π‘’(π‘₯𝑝) β‰  π‘π‘œπ‘‘π‘’(π‘₯π‘ž). So, we get the upper bound of the irregular chromatic number on the pan graph is πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ π‘˜ + 1. The lower and upper bound of the irregular chromatic number of pan graph is π‘˜ +1 ≀ πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) ≀ π‘˜ + 1. So, πœ’π‘–π‘Ÿ(π‘ƒπ‘Žπ‘›) = π‘˜ + 1. Theorem 4. The irregular chromatic number of sun graph 𝑺𝒏 for π’Œ 𝟐 + π’Œ+ 𝟏 ≀ 𝒏 ≀ π’ŒπŸ + πŸ‘π’Œ+ 𝟐 and π’Œ ∈ 𝑡 is π’Œ+ 𝟐. Proof: This graph has the vertex set 𝑉(𝑆𝑛) = {π‘₯𝑖|1 ≀ 𝑖 ≀ 𝑛} βˆͺ {𝑦𝑖|1 ≀ 𝑖 ≀ 𝑛} and the edge set is 𝐸(𝑆𝑛) = {π‘₯𝑖π‘₯𝑖+1|1 ≀ 𝑖 ≀ 𝑛 βˆ’1} βˆͺ{π‘₯1π‘₯𝑛} βˆͺ {π‘₯𝑖𝑦𝑗|1 ≀ 𝑖 ≀ 𝑛,1 ≀ 𝑗 ≀ 𝑛}. |𝑉(𝑆𝑛)| = 2𝑛 and |𝐸(𝑆𝑛)| = 2𝑛. We need to prove upper bound πœ’π‘–π‘Ÿ(𝑆𝑛) ≀ π‘˜ + 2 and lower bound πœ’π‘–π‘Ÿ(𝑆𝑛) β‰₯ π‘˜ + 2. First, we prove that the lower bound of the irregular chromatic number on the sun graph is πœ’π‘–π‘Ÿ(𝑆𝑛) β‰₯ π‘˜ + 2. Assume that πœ’π‘–π‘Ÿ(𝑆𝑛) < π‘˜ + 2. If π‘₯𝑖 colored with k+1-color, then at most k + 1 vertex of the same color. Because of π‘₯π‘˜π‘¦π‘˜ ∈ 𝐸(𝑆𝑛) and π‘¦π‘˜ are colored with k+1-color, then there are two neighboring vertices having the same color, namely 𝑐(π‘₯π‘˜) = 𝑐(π‘¦π‘˜). This contradicts the definition of proper coloring, where each neighboring vertex must have a different color. Thus, it can be concluded that the upper bound of the irregular chromatic number of a sun graph is πœ’π‘–π‘Ÿ(𝑆𝑛) β‰₯ π‘˜ + 2. Next, we prove that the upper bound of the irregular chromatic number on the sun graph is πœ’π‘–π‘Ÿ(𝑆𝑛) ≀ π‘˜ + 2. The color function c on this graph is defined as follows. On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana 509 ο‚· For π‘₯𝑖 vertices, where 𝑛 ≑ 1(π‘šπ‘œπ‘‘ π‘˜ + 2) the color of the vertices 1,2,3,…,π‘˜ + 2,1,2,3,…,π‘˜ + 2 periodically. ο‚· For π‘₯𝑖 vertices, where n not equivalent 1(π‘šπ‘œπ‘‘ π‘˜ + 2) the color of the vertices 1,2,3,…,π‘˜ + 2,1,2,3,…,π‘˜ + 2,…,1,2,3,…,π‘˜ + 2,2 periodically. ο‚· For 𝑦𝑖 vertices, 𝑐(π‘₯𝑖) β‰  𝑐(𝑦𝑗) where π‘₯𝑖𝑦𝑗 ∈ 𝐸(𝑆𝑛), and 𝑐(π‘₯𝑖) β‰  𝑐(π‘₯𝑗) for π‘₯π‘–π‘¦π‘˜,π‘₯𝑗𝑦𝑙 ∈ 𝐸(𝑆𝑛), with 𝑐(π‘¦π‘˜) = 𝑐(𝑦𝑙),π‘˜ β‰  𝑙. We know that the π‘π‘œπ‘‘π‘’(π‘₯𝑝) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜) and π‘π‘œπ‘‘π‘’(π‘₯π‘ž) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜), then based on the color function obtained, π‘π‘œπ‘‘π‘’(π‘₯𝑝) β‰  π‘π‘œπ‘‘π‘’(π‘₯π‘ž). Thus, the upper bound of the irregular chromatic number on a sun graph is πœ’π‘–π‘Ÿ(𝑆𝑛) ≀ π‘˜ + 2. The ilustration of irregular coloring of sun graph can be seen in Figure 2. The lower and upper bound of the irregular chromatic number of caveman graph is π‘˜ + 2 ≀ πœ’π‘–π‘Ÿ(𝑆𝑛) ≀ π‘˜ + 2. So, πœ’π‘–π‘Ÿ(𝑆𝑛) = π‘˜ + 2. Figure 2. πœ’π‘–π‘Ÿ(𝑆𝑛) = 4 Theorem 5. Irregular chromatic number of peach graph π‘ͺπ’Ž π’Ž for π’Ž β‰₯ πŸ‘ is π’Ž + 𝟏. Proof: This graph has the vertex set 𝑉(πΆπ‘š π‘š) = {π‘₯𝑖|1 ≀ 𝑖 ≀ π‘š}βˆͺ{𝑦𝑖|1 ≀ 𝑖 ≀ π‘š} and the edge set is 𝐸(πΆπ‘š π‘š) = {π‘₯𝑖π‘₯𝑖+1|1 ≀ 𝑖 ≀ π‘š βˆ’ 1} βˆͺ{π‘₯1π‘₯π‘š} βˆͺ{π‘₯1𝑦𝑖|1 ≀ 𝑖 ≀ π‘š}. |𝑉(πΆπ‘š π‘š)| = 2π‘š and |𝐸(πΆπ‘š π‘š)| = 2π‘š. We need to prove upper bound πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) ≀ π‘š + 1 and lower bound πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) β‰₯ π‘š +1. First, we prove that the lower bound of the irregular chromatic number on the peach graph is πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) β‰₯ π‘š + 1. Assume that πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) < π‘š + 1. We have the following cases. Case 1. 𝑐(π‘₯𝑖) = 𝑐(𝑦𝑗), π‘₯𝑖𝑦𝑗 ∈ 𝐸(πΆπ‘š π‘š). It’s contradicts. Case 2. 𝑐(π‘₯π‘˜) = 𝑐(𝑦𝑙), π‘˜ β‰  𝑙,π‘¦π‘˜π‘₯1,𝑦𝑗π‘₯1 ∈ 𝐸(πΆπ‘š π‘š). It’s contradicts. Based on case 1 and case 2 above, the lower bound is obtained from the irregular chromatic number on the peach graph is πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) β‰₯ π‘š + 1. Next, we prove that the upper bound of the irregular chromatic number on the peach graph is πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) ≀ π‘š + 1. The color function c on this graph is defined as follows. 𝑐(π‘₯𝑖) = 𝑖,1 ≀ 𝑖 ≀ π‘š 𝑐(𝑦𝑗) = { 𝑗 π‘“π‘œπ‘Ÿ 2 ≀ 𝑗 ≀ π‘š π‘š + 1 π‘“π‘œπ‘Ÿ 𝑗 = 1 On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana 510 Based on the color function above, the code for each vertex in the peach graph is obtained as follows. π‘π‘œπ‘‘π‘’ (π‘₯𝑖) = { (𝑐(π‘₯𝑖),0,2,1,1,1,1,0,0,…,1,1,2⏟ π‘š+1 π‘“π‘œπ‘Ÿ 𝑖 = 1 (𝑐(π‘₯𝑖),0,…,0⏟ , π‘–βˆ’2 1,0,1,0,…,0⏟ , π‘šβˆ’1 π‘“π‘œπ‘Ÿ 2 ≀ 𝑖 ≀ π‘š βˆ’ 1 (𝑐(π‘₯𝑖),1,0,0,0,0,0,0,…,0,0,1,0⏟ π‘š+1 π‘“π‘œπ‘Ÿ 𝑖 = π‘š π‘π‘œπ‘‘π‘’(𝑦𝑖) = (𝑐(𝑦𝑖),1,0,0,0,…,0,0,0)⏟ π‘š+1 Based on the color function and vertex code obtained, each neighboring vertex has a different color and each vertex in the graph has a different color code. Therefore, the upper bound of the irregular chromatic number on the peach graph is πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) ≀ π‘š + 1. The illustration of irregular coloring of sun graph can be seen in Figure 3. The lower and upper bound of the irregular chromatic number of peach graph is π‘š + 1 ≀ πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) ≀ π‘š + 1. So, πœ’π‘–π‘Ÿ(πΆπ‘š π‘š) = π‘š + 1. Figure 3. πœ’π‘–π‘Ÿ(𝐢8 8) = 9 Theorem 6. Irregular chromatic number of caveman graph π‘²π’Ž,πŸ‘ for π’Œ 𝟐 + π’Œ + 𝟏 ≀ π’Ž ≀ π’ŒπŸ + πŸ‘π’Œ+ 𝟐 and π’Œ ∈ 𝑡 is π’Œ+ 𝟐. Proof: This graph has the vertex set 𝑉(πΎπ‘š,3) = {π‘₯𝑖|1 ≀ 𝑖 ≀ 2π‘š} βˆͺ{𝑦𝑗|1 ≀ 𝑗 ≀ π‘š} and the edge set is 𝐸(πΎπ‘š,3) = {π‘₯𝑖π‘₯𝑖+1|1 ≀ 𝑖 ≀ 2π‘š βˆ’ 1} βˆͺ{π‘₯1π‘₯2π‘š}βˆͺ {π‘₯𝑖𝑦𝑖|1 ≀ 𝑖 ≀ 2π‘š,𝑖 = π‘œπ‘‘π‘‘,1 ≀ 𝑗 ≀ π‘š}. |𝑉(πΎπ‘š,3)| = 3π‘š + 6 and |𝐸(πΎπ‘š,3)| = 3π‘š + 6. We need to prove upper bound πœ’π‘–π‘Ÿ(πΎπ‘š,3) ≀ π‘˜ + 2 and lower bound πœ’π‘–π‘Ÿ(πΎπ‘š,3) β‰₯ π‘˜ + 2. First, we prove that the lower bound of the irregular chromatic number on the caveman graph is πœ’π‘–π‘Ÿ(πΎπ‘š,3) β‰₯ π‘˜ + 2. Assume that πœ’π‘–π‘Ÿ(πΎπ‘š,3) ≀ π‘˜ + 2, for example is πœ’π‘–π‘Ÿ(πΎπ‘š,3) = π‘˜ + 1. If π‘₯𝑖, where deg(π‘₯𝑖) = 3 is colored with k+1-color, then at most k+1 vertex of the same color. Because of π‘₯𝑖𝑦𝑗 ∈ 𝐸(πΎπ‘š,3) and 𝑦𝑗 are colored with k+1-color, then there are two neighboring vertices having the same color, namely 𝑐(π‘₯𝑖) = 𝑐(𝑦𝑗). This contradicts the definition of proper coloring, where each neighboring vertex must have a different color. On Irregular Colorings of Unicyclic Graph Family Arika Indah Kristiana 511 Thus, it can be concluded that the lower bound of the irregular chromatic number of a caveman graph is πœ’π‘–π‘Ÿ(πΎπ‘š,3) β‰₯ π‘˜ + 2. Next, we prove that the upper bound of the irregular chromatic number on the caveman graph is πœ’π‘–π‘Ÿ(πΎπ‘š,3) ≀ π‘˜ + 2. The color function c on this graph is defined as follows. ο‚· For π‘₯𝑖 vertices, where deg(π‘₯𝑖) = 3, the color of the vertices 1,2,3,…,π‘˜ + 2,1,2,3,…,π‘˜ + 2 periodically. ο‚· For π‘₯𝑖 and π‘₯𝑗 vertices, with deg(π‘₯𝑖) = deg(π‘₯𝑗) = 2, π‘₯𝑖 = π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1and π‘₯𝑗 = π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1. If 𝑆 = {1,2,3,…,π‘˜ + 2}, π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1,π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1 ∈ 𝑉(πΎπ‘š,3) β‰₯ π‘˜ + 1, with 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) ∈ 𝑆. If 𝑐(π‘₯(𝑝(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(𝑝+2(π‘šπ‘œπ‘‘ 𝑛))+1) β‰  𝑐(π‘₯(π‘ž(π‘šπ‘œπ‘‘ 𝑛))+1),𝑐(π‘₯(π‘ž+2(π‘šπ‘œπ‘‘ 𝑛))+1), then 𝑐(π‘₯(𝑝+1(π‘šπ‘œπ‘‘ 𝑛))+1) = 𝑐(π‘₯(π‘ž+1(π‘šπ‘œπ‘‘ 𝑛))+1). ο‚· For 𝑦𝑗 vertices, 𝑐(π‘₯𝑖) β‰  𝑐(𝑦𝑗) for π‘₯𝑖𝑦𝑗 ∈ 𝐸(πΎπ‘š,3) ο‚· For 𝑦𝑗 vertices, 𝑐(π‘¦π‘˜) = 𝑐(𝑦𝑙) when π‘₯𝑖𝑦𝑗,π‘₯𝑗𝑦𝑙 ∈ 𝐸(πΎπ‘š,3) and 𝑐(π‘₯𝑖) β‰  𝑐(𝑦𝑗) We know that the π‘π‘œπ‘‘π‘’(π‘₯𝑝) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜) and π‘π‘œπ‘‘π‘’(π‘₯π‘ž) = (π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘˜) then based on the color function obtained, π‘π‘œπ‘‘π‘’(π‘₯𝑝) β‰  π‘π‘œπ‘‘π‘’(π‘₯π‘ž). Thus, the upper bound of the irregular chromatic number on a caveman graph is πœ’π‘–π‘Ÿ(πΎπ‘š,3) ≀ π‘˜ + 2. The illustration of irregular coloring of caveman graph can be seen in Figure 4. The lower and upper bound of the irregular chromatic number of peach graph is π‘˜ + 2 ≀ πœ’π‘–π‘Ÿ(πΎπ‘š,3) ≀ π‘˜ + 2. So, πœ’π‘–π‘Ÿ(πΎπ‘š,3) = π‘˜ + 2. 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