Coloring Sums of Extensions of Certain Graphs

Recall that the minimum number of colors that allow a proper coloring of graph $G$ is called the chromatic number of $G$ and denoted by $\chi(G).$ In this paper the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum are introduced. The extended graph $G^x$ of a graph $G$ was recently introduced for certain regular graphs. We further the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum to extended paths and cycles. The paper concludes with \emph{patterned structured} graphs.


Introduction
For general notation and concepts in graph and digraph theory, we refer to [2,4,5]. Unless mentioned otherwise, all graphs mentioned in this paper are simple, connected, finite and undirected graphs of order n ≥ 2. We recall that the minimum number of colors that allow a proper coloring of graph G is called the chromatic number of G and denoted χ(G). Since coloring may be effected from any alphabet like in an application of cryptographical coding of vertices, we use a preferred definition slightly different from the contemporary definition found in the literature. Consider a proper k-coloring of a graph G and denote the set of k colors, C = {c 1 , c 2 , c 3 , . . . , c k }. Also consider the disjoint subsets of V (G) i.e. V c i = {v j : v j → c i , v j ∈ V (G), c i ∈ C}, 1 ≤ i ≤ k. Clearly, V c i . If for largest k ∈ N a proper k-coloring is found such that there exist v s ∈ V c i and v t ∈ V c j , such that edge v s v t ∈ E(G) for all distinct pairs of colors c i , c j then the b-chromatic number of G is defined to be ϕ(G) = k. Such a coloring is called a b-coloring of G.
In a paper by Anirban Banerjee and Saptarshi Bej [1] the concept of extending regular graphs has been introduced. We shall use the idea of extending a graph (also called an extended graph) in general and study how certain interesting invariants change in the extended graph.

On Extensions of Graphs
We formally define the notion of a degree-extension of a graph G.
Definition 2.1. (i) Let G be a graph of even order n ≥ 2. If the complement graph G c has a perfect matching say M, then G x c = G + M is called a complete degree-extension of G, also called an c-extended G.
(ii) Let G be a graph of odd order n ≥ 3 and let u ∈ V (G) such that d G (u) = ∆(G). If the complement graph (G−u) c has a perfect matching say M , then G x ic = G+M is called an incomplete degree-extension of G, also called an ic-extended G.
Note that the choice of vertex u, d G (u) = ∆(G) in Definition 2.1(ii) is by default. Other criteria can apply of which u, d G (u) = δ(G) is an obvious other criteria. An immediate and important result following from Definition 2.1 is given by the next corollary.
In [1] it is shown that a r-regular graph G of even order n with r < n 2 always has a degreeextension G x c . It means that G can always be extended to a (r + 1)-regular graph G x c by adding edges to G. First, we generalise the result to any simple connected graph G for which ∆(G) < n 2 . Theorem 2.2. Let G be a graph of order n with ∆(G) < n 2 , then graph G has a complete degreeextension G x c .
Proof. Following from Definition 2.1 we consider two cases. Case (i): See [1]. Assume n is even. Since ∆(G) < n 2 the complement graph G c has δ(G c ) ≥ n 2 . Recalling Dirac's Theorem it follows that G c is Hamiltonian. So for each Hamilton cycle in G c exactly two perfect matchings exist in G c . Therefore, G has at least two complete degree-extensions, It is observed that ⌊ n 2 ⌋ edges must be added to obtain an incomplete degree-extension G x ic . Henceforth we will only consider a graph G of even order n.
If n is even and if k distinct Hamilton cycles exist in G c then 2k complete extended graphs exist. For graphs on even number of vertices we have the next result. Proof. If the complement graph G c has a spanning path of length n−1 exactly one perfect matching corresponding to that path in G c exists. The result follows immediately. Proof. Denote the spanning path in V i (G) c byP (i) , 1 ≤ i ≤ ℓ. From each P (i) we select the unique Clearly the smallest path for which Theorem 2.2 finds application is P 4 . Let the vertices of P 4 consecutively be labeled . For P 6 we find the application of Theorem 2.2 allows P x 1 6,c = P 6

χ ′ -Chromatic Sum and χ + -Chromatic Sum of Certain Graphs
We recall that a vertex coloring of a graph G such that adjacent vertices are not allocated the same color is called a proper coloring of G. The minimum number of colors in a proper coloring of G is called the chromatic number, χ(G).

3.1
The χ ′ -Chromatic Sum and χ + -Chromatic sum of Extended Paths and Extended Cycles In a paper [9], Lisna and Sunitha introduce a new concept called the b-Chromatic Sum of a Graph.
As stated in [7] there are k! ways of allocating the colors to the vertices of G. Let the color weight θ(c i ) be the number of times a color c i is allocated to vertices. In general we refer to the color sum of a coloring S and define it,

b-chromatic sum define by Lisna and Sunitha is given by
G}. This interesting new invariant motivates similar concepts in graph coloring. Also see [7].
For a graph G the χ ′ -chromatic sum is defined to be: For a graph G the χ + -chromatic sum is defined to be: Further motivation for these new invariants is as follows. If the colors represent different technology types and the configuration requirement is that at least one unit per technology type must be placed at a point in a network without similar technology types being adjacent, two further considerations come into play. Firstly, if the higher indexed colors represent technology types with higher failure rate (risk) then the placement of the maximal number of higher indexed units is the solution to ensure a functional network. On the other hand is the lower indexed colors represent a less costly (procurement, installation, commissioning and maintenance) technology type, and minimising total cost is the priority, then the placement of maximal number of lower indexed units is the desired solution. We recall two important results from [7].
For a path P n , n ≥ 1 the χ ′ -chromatic sum and χ + -chromatic sum are given by: For a cycle C n the χ ′ -chromatic sum and χ + -chromatic sum are given by: . Also we note that P x 4 6 , (by symmetry P x 5 6 as well) allows a minimum proper coloring Up to isomorphism P 6 has 4 distinct complete extensions. These observations motivate the next definitions. Definition 3.3. For a graph G of even order n, such that the complement graph G c has at least one perfect matching then, if, up to isomorphism the graph G has G Definition 3.4. For a graph G of even order n, such that the complement graph G c has at least one perfect matching then, if up to isomorphism the graph G has G Theorem 3.3. For a path n , n ≥ 4 and even, we have: . Now interchange the colors c 1 and c 3 . It follows that χ + (P x 4,c ) = 9. Case (ii)(a): Consider P n , n ≥ 6. Subcase (ii)(a)(1): If n 2 is odd, consider P x n,c = P n + Since equal number of vertices are left with degree 2 and color c 3 ,c 2 respectively it is always possible to complete the extension to obtain P x n,c having the allocated minimum proper coloring. Clearly θ(c 1 ) = 1, θ(c 2 ) = t − 1 and θ(c 3 ) = t. (2): The result follows similar to Subcase (ii)(b)(1).

On Bipartite Graphs
It is well known that a graph G is bipartite if and only G contains no odd cycle. Up to equivalence a graph (connected) which contains no odd cycle has a unique vertex-set partition say X, Y such that if v, u ∈ X then vu / ∈ E(G) and similarly for vertices in Y . If |X| and |Y | are even we say the partition is even balanced, else if |X| and |Y | are odd it is odd balanced. We also say ordered pairs  Assume there are ℓ such bi-distinct pairs. Subcase 1(i): If ℓ is even, add the edges v i u j for all ℓ bi-distinct pairs. Clearly n − ℓ vertices in X can be paired into bi-distinct vertex pairs. Add those corresponding pairwise edges. Similarly, n − ℓ vertices in Y can be paired into bi-distinct vertex pairs. Add those corresponding pairwise edges as well. The new graph obtained through this construction is a complete degree-extension G x c of G. Subcase 1(ii): If ℓ is odd select any ℓ − 1 bi-distinct vertex pairs. Construct a new graph similar to Case 1(i) which is clearly G x c . Case 2: Consider the odd balanced bipartite graph B n,m which is isomorphic to G. As before assume |X| = n ≥ m = |Y | and label the vertices similarly. Subcase 2(i): If If ℓ is even select any ℓ − 1 bi-distinct vertex pairs. Construct a new graph similar to Case 1(i) which is clearly G x c . Subcase 2(ii): If ℓ is even, construct a new graph similar to Case 1(ii) which is clearly G x c .
We note that in all cases above the subgraph G − {v i , u j : ∀ℓ or ℓ − 1, (v i , u j ) is a bi-distinct non-adjacent pair of vertices} is a complete bipartite subgraph. Theorem 3.6. If G is isomorphic to a balanced bipartite graph B n,m with |X| = n ≥ m = |Y | and G has a maximum ℓ bi-distinct pairs of vertices (v i , u j ), v i ∈ X, u j ∈ Y , then: Case 1: If both n, m are even: Case 2: If both n, m are odd: Proof. Case 1: Assume both |X| and |Y | are even. Label the vertices in X, v 1 , v 2 , v 3 . . . , v n and those in Y , u 1 , u 2 , u 3 , . . . , u m . Subcase 1(i): Let ℓ be even. Without loss of generality label the vertices of X, Y which belong to the ℓ bi-distinct non-adjacent pairs as v 1 , v 2 , v 3 , . . . , v ℓ and u 1 , u 2 , u 3 , . . . , u ℓ , respectively such that (v i , u j ), 1 ≤ i ≤ ℓ are non-adjacent. Add edges v i u i , 1 ≤ i ≤ ℓ and assign the colors Label the rest of the vertices of X and Y to be v ℓ+1 , v ℓ+2 . . . , v n and u ℓ+1 , u ℓ+2 , . . . , u m , respectively. Now add the edges v ℓ+1 v ℓ+2 , u ℓ+1 u ℓ+2 and assign the colors Note that the maximum matching in a bipartite graph hence, the maximum bi-distinct nonadjacent pairs of vertices and the value ℓ can be determined by amongst others, the n 5 2 -Algorithm described by Hopcroft and Karp in [6]. We now discuss a special case of this theorem where |X| = |Y |, which means both the partitions of the bipartite graph have equal cardinality. This class of graph inclues all regular bipartite graphs. For this case we calculate both χ ′ (G x ) and χ + (G x c ).
Definition 3.5. Let G be a bipartite graph, with vertex partitions X an Y . Let S be a subset of X. For, v ∈ S we define N c (v) to be the set of vertices in Y that are not adjacent to v. We define N c (S) to be v∈S N c (v).
Corollary 3.7. Let G be a bipartite graph of even order n, with vertex partitions X an Y , such that |X| = |Y |. If for every S ⊆ X, |N c (S)| ≥ |S| then, Proof. Case (i): Note that in G, X and Y are independent sets. Hence, in G c vertices in X and Y will be cliques of size |X| and |Y |, respectively. Let us denote these cliques as K |X| and K |Y | respectively. Also, note that the graph G c − E(K |X| ) − E(K |Y | ) is bipartite. By Hall's Theorem the graph G c −E(K |X| )−E(K |Y | ) has a perfect matching M , since in G, for every S ⊆ X, |N c (S)| ≥ |S|. Thus, G can always be extended to a bipartite graph. Hence, χ ′ (G x c ) = 3n 2 . Case (ii): Now, let us assign a different extension of G. Choose two arbitrary edges from M , x 1 y 1 and x 2 y 2 , such that x 1 , x 2 ∈ X and y 1 , y 2 ∈ Y . Now, for extending G, use the edges M − x 1 y 1 − x 2 y 2 + x 1 x 2 + y 1 y 2 . This can always be done since the graph is bipartite. Now, we can color x 1 and y 1 using c 1 . Vertices in X − x 1 and Y − y 1 can be colored using c 2 and c 3 , respectively. This obviously shows that, χ + (G x c ) = 5n 2 − 3.

On some specific classes of Graphs
Lemma 4.1. Let G be a graph of order n, such that δ(G) > n 2 , then diam(G) ≤ 2.
Proof. Let A be the adjacency matrix of G. Every entry a i,j of A 2 is the scalar product of the i-th row and the j-th column of A. Since every row and column of A has at least n 2 entries equal to 1, every a i,j > 0, a i,j an entry of A 2 . Hence for any two distinct vertices v, u ∈ V (G) we have sharing a common edge say uv. If in a minimum proper coloring of H 1 the vertex coloring is v → c i , u → c j then after applying Lemma 4.3 the χ-number of the partial H-gridlike element remains the same. Case (ii): Consider a H-cloverlike element and without loss of generality assume the common vertex is v. If in a minimum proper coloring of H 1 the vertex coloring is v → c i then after applying Lemma 4.3 (i) the χ-number in the partial H-cloverlike element remains the same. By iteratively applying Lemma 4.3 (i) to all pairs, the χ-number of the whole H-cloverlike element remains the same. Case (iii): Consider a H-booklike element. Similar reasoning as in Case (i) follows. Case (iv): First consider adjacent H-elements H 1 and H 2 as the vertex join between H 1 and an end vertex of a path and apply Lemma 4.3 (i). Thereafter, consider the vertex join between the other end vertex of the path and H 2 and apply Lemma 4.3 (i) again. Clearly the χ-number remains the same.
Invoking Cases (i) to (iv) throughout a H-treelike graph G * settles the result in general.

Conclusion
We recall that an almost regular graph G is such that ∆(G) − δ(G) = 1. A partial extension of an almost regular graph G is defined to be the graph G px obtained by adding edges to G such that G px is ∆(G)-regular. A path is such an almost regular graph so it is easy to see that P px n = C n , therefore χ ′ (P px n ) = χ ′ (C n ) and χ + (P px n ) = χ + (C n ). There is scope to research χ ′ (G px n ) and χ + (G px n ) for other classes of almost regular graphs.
Kouider and El Sahili [7] showed that for a r-regular graph G on at least r 4 vertices the b-chromatic number is ϕ(G) = r + 1. In a paper by Cabello and Jakovac [3] they improved the result by bounding the result to graphs having at least 2r 3 vertices. So it is easy to see that for C n , n ≥ 54 we have ϕ(C n ) = 3 and ϕ(C x n,c ) = 4. In general it will be worthy to research the relationship, if any, between ϕ ′ (C n ) and ϕ ′ (C x n,c ) as well as that between ϕ + (C n ) and ϕ + (C x n,c ) and perhaps for other classes of regular graphs. and insightfull comments. The authors also wish Professor Banerjee the complete return of great health. We need his amazing intellectual capacity to guide many of us for many years to come.
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