Total Colorings of Some Classes of FourRegular Circulant Graphs

The total chromatic number, $\chi''(G)$ is the minimum number of colors which need to be assigned to obtain a total coloring of the graph $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing that for any graph, $\chi''(G) \leq \Delta(G)+2 $, where $\Delta(G)$ represents the maximum degree of $G$. In this paper we obtained the total chromatic number for some classes of four regular circulant graphs.


Introduction
Let G be a simple graph with vertex set V (G) and edge set E(G).The total coloring of a graph G is an assignment of colors to vertices and edges such that no two adjacent vertices or edges or edges incident to a vertex receives a same color.The total chromatic number of a graph G, denoted by χ ′′ (G), is the minimum number of colors required for its total coloring.It is clear that χ ′′ (G) ≥ ∆(G)+1, where ∆(G) is the maximum degree of G.
Behzad [1] and Vizing [9] have independently proposed the Total Coloring Conjecture (TCC) which states that any simple graph G, χ ′′ (G) ≤ ∆(G) + 2. The graphs that can be totally colored by at-least ∆(G) + 1 colors are said to be Type I graphs whereas the graphs which can be colored by ∆(G) + 2 colors are said to be Type II graphs.The decidability algorithm for total coloring is NP-complete even for cubic bipartite graph [8].Good survey of techniques and other results on total coloring can be found in Yap [10], Borodin [2] and Geetha et al. [4].In this paper, we obtain the total chromatic number of some four regular circulant graphs are Type I.

Four Regular Circulant Graphs
For a sequence of positive integers 1 Campos and de Mello [3] proved that C 2 n , n = 7, are Type I and C 2 7 is Type II.We know that K 4,4 is a four regular circulant graph and it is Type II [10].A Unitary Cayley graph is a circulant graph with vertex set V (G) = {0, 1, 2, ..., n − 1} and two vertices x and y are adjacent if and only if gcd((x − y), n) = 1.Prajnanaswaroopa et al. [7], proved that al most all Unitary Cayley graphs of even order are Type I and odd order satisfies TCC.In this paper, we considered the four regular circulant graphs of the form Mauro and Diana [5] proved that the graphs C n (2k, 3) are Type I for n = (8µ + 6λ)k, with k ≥ 1 and non-negative integers µ and λ.Riadh Khennoufa and Olivier Togni [6] studied total colorings of circulant graphs and proved that every 4-regular circulant graphs C 6p (1, k), p ≥ 3 and k < 3p with k ≡ 1 mod 3 or k ≡ 2 mod 3, are Type I. Other cases are still open.Also they proved that the total chromatic number of C 5p (1, k), with k < 5p 2 , k ≡ 2 mod 5, k ≡ 3 mod 5 is 5.In the following theorem, we prove that the graphs C 5p (1, k) are Type I for the remaining cases k ≡ 1 mod 5 and k ≡ 4 mod 5. Proof.Let q 1 = gcd(5p, a) and q 2 = gcd(5p, b).The circulant graphs G = C 5p (a, b) are four regular graphs with q 1 cycles of order 5p q 1 and q 2 cycles of order 5p Since, the all the four graphs C 5p (a, b), C 5p (b, a), C 5p (n − a, b) and C 5p (a, n − b) are isomorphic to each other, for the edge colorings, we need to consider the three cases.
Case 1: a ≡ 1 mod 5 and b ≡ 1 mod 5.The edges of cycles can be colored by setting ϕ(v i v (i+a) mod 5 ) = (i + 2) mod 5 and Case 2: a ≡ 2 mod 5 and b ≡ 2 mod 5.The edges of cycles can be colored by setting ϕ(v i v (i+a) mod 5 ) = (i + 3) mod 5 and Case 3: a ≡ 1 mod 5 and b ≡ 2 mod 5.The edges of cycles can be colored by setting ϕ(v i v (i+a) mod 5 ) = (i + 3) mod 5 and Therefore, five colors are used for totally color the graph.
In the following theorem, we prove some classes of four regular circulant graphs C n (a, b) of order 3p are Type I. Proof.Let q 1 = gcd(3p, a) and q 2 = gcd(3p, b).The circulant graphs G = C 3p (a, b) are four regular graphs with q 1 cycles of order 3p q 1 and q 2 cycles of order 3p q 2 .Let ϕ : V (G) ∪ E(G) → {0, 1, 2, 3, 4} be a mapping obtained by the following process.
Let C i be the cycles of order 3p q 2 with the vertices v ia , 0 ≤ i ≤ q 2 − 1.First we consider the cycle C 0 .If q 2 = 1 then the vertices and edges of C 0 are colored with the colors 0, 3 and 1 cyclically, starting with v 0 receiving the color 0. Otherwise the vertices and edges of C 0 are colored with the colors 3, 1 and 0 cyclically, starting with v 0 receiving the color 1.Now, consider the cycle C 1 .The vertices and edges of C 1 are colored with the colors 1, 0 and 4 cyclically, starting with v a receiving the color 1.For the cycles C i , 2 ≤ i ≤ q 2 − 1, if i is even then the vertices and edges of C i are colored with the colors 0, 2 and 1 cyclically, starting with v ia receiving the color 0 and i is odd they are colored with the colors 1, 0 and 2 cyclically, starting with v ia receiving the color.
The edges of cycles of order 3p q 1 are colored in the following way: if vertex v i ∈ C 0 then ϕ(v i v i+a ) = 2, if v i ∈ C i where i is odd then ϕ(v i v i+a ) = 3 and if v i ∈ C i where i is even then ϕ(v i v i+a ) = 4. Therefore, only five colors are used for totally coloring the graph.Hence, ϕ is a Type I coloring of G For the circulant graphs C n (a, b) where n = 3p and p is odd, which we considered in the above theorem, the value of b is restricted to factors and multiple of p.In the following theorem, we consider few classes of circulant graphs C n (1, k) where n = 9p are Type I.
Case 1: p is even.
When p is even, 9p will be a multiple of 6. Riadh Khennoufa and Olivier Togni [6] proved that the total chromatic number of G = C 6p (1, k), with k < 5p 2 , k ≡ 1 mod 3, k ≡ 2 mod 3 is 5. From this, one can easily see that the circulant graph C 9p (1, k), where p ≥ 1 and k < 9p 2 , k ≡ 1 mod 3, k ≡ 2 mod 3 are Type I. Now, we consider the remaining case, k ≡ 0 mod 3. The vertices v i are colored by ϕ(v i ) = (i mod 3 + ⌊ i k ⌋ mod 3) mod 3 if q = k, else the vertices v i are colored by ϕ(v i ) = (2i mod 3 − ⌊ i 3 ⌋ mod 3) mod 3. The edges of the internal cycles can be colored by setting ϕ(v i v (i+k) mod 9p ) = (2ϕ(v (i+k) mod 9p ) − ϕ(v i )) mod 3.In this coloring process, the vertices and the internal edges of G are colored using only three colors 0, 1 and 2. Now, the edges of the outer cycle can be colored with two colors 3 and 4 as it is of even order.Therefore, five colors are used for total coloring the graph, hence the graph C 9p (1, k) is Type I. Case 2: p is odd.
The case when q = 1, follows from Theorem 2.2.Now, we consider the case when q = 1.Sub case 2.1: k ≡ 1 mod 9 The vertices v i where 0 ⌋.The edges of the internal cycles can be colored by setting if i ≡ ).The colors used for vertex v i and edges incident to it is given in the table below as an ordered triplet (ϕ x ≡ i mod 9 The common missing color for vertices v i and v i+1 can be used for coloring the edge v i v i+1 .Therefore, five colors are used for totally coloring the graph, hence G = C 9p (1, k) Type I.
The vertices v i where 0 ⌋.The edges of the internal cycles can be colored by setting ϕ(v i v (i+k) mod 9p ) = ϕ(v (i+k+1) mod 9p ).The colors used for vertex v i and edges incident to it is given in the table below as an ordered triplet (ϕ x ≡ i mod 9 The common missing color for vertices v i and v i+1 can be used for coloring the edge v i v i+1 .Therefore, five colors are used for the total coloring the graph, hence G = C 9p (1, k) is Type I.
In Theorem 2.2, we considered few classes of circulant graph C n (a, b) of order n = 3p, where p is an odd integer.Now, in the following theorem, we consider few classes of four regular circulant graphs C n (a, b) of order n = 6p.Proof.Let q 1 = gcd(6p, a) and q 2 = gcd(6p, b).The circulant graphs G = C 6p (a, b) are four regular graphs with q 1 cycles of order 6p q 1 and q 2 cycles of order 6p q 2 .The circulant graphs considered in the hypothesis can be colored in a similar manner irrespective of p being odd or even, if 6p q 1 is odd we swap the value of a and b, as graph C n (a, b) is isomorphic to C n (b, a).Let ϕ : V (G) ∪ E(G) → {0, 1, 2, 3, 4} be a mapping.The vertices v i are colored by ϕ(v i ) = i mod 3 and the edges of cycles of order 6p q 2 be colored by setting ϕ(v i v (i+a) mod 6p ) = (2ϕ(v (i+a) mod 3p ) − ϕ(v i )) mod 3. Now, the edges of cycle 6p q 1 with two colors 3 and 4 as it is a cycle with even order.Therefore, five colors are used for the total coloring ϕ of the graph, hence graph G is Type I.

Theorem 2 . 4 .
Every circulant graph C 6p (a, b) where a, b ≡ 0 mod 3 is Type I, if p is even.Also, C 6p (a, b) where a, b ≡ 0 mod 3 is Type I, if p is odd and gcd(a, b) = 1.
1, 2, ..., n − 1} and two vertices x and y are adjacent if x ≡ (y ± d i ) mod n for some i where 1 ≤ i ≤ l.A power of cycles graph C k n is a graph with vertex set V (G) = {0, 1, 2, ..., n − 1} and two vertices x and y are adjacent if and only if |x − y| ≤ k.It is easy to see that the four regular circulant graph