Some New Results on Proper Colouring of Edge-set Graphs

In this paper, we present a foundation study for proper colouring of edge-set graphs. The authors consider that a detailed study of the colouring of edge-set graphs corresponding to the family of paths is best suitable for such foundation study. The main result is deriving the chromatic number of the edge-set graph of a path, $P_{n+1}$, $n \geq 1$. It is also shown that edge-set graphs for paths are perfect graphs.


INTRODUCTION
For general notation and concepts in graphs and digraphs see [1,2,12]. Unless mentioned otherwise, all graphs we consider in this paper are finite, simple, connected and undirected graphs.
For a set of distinct colours C = {c 1 , c 2 , c 3 , . . . , c }, a vertex colouring of a graph G is an assignment ϕ : V (G) → C . A vertex colouring is said to be a proper vertex colouring of a graph G if no two distinct adjacent vertices have the same colour. The cardinality of a minimum set of colours in a proper vertex colouring of G is called the chromatic number of G and is denoted r − χ (G) and r + χ (G), have been defined as the minimum value and maximum value of r χ (G) over all permissible colour allocations. If we relax connectedness, it follows that the null graph N n of order n ≥ 1 has r − (N n ) = r + (N n ) = n. For bipartite graphs and complete graphs, K n it follows that, r − (G) = r + (G) = n and r − (K n ) = r + (K n ) = n.
We observe that if it is possible to permit a chromatic colouring of any graph G of order n such that the star subgraph obtained from vertex v as center and its open neighbourhood N(v) the pendant vertices, has at least one coloured vertex from each colour for all v ∈ V (G) then r χ (G) = n. Certainly, examining this property for any given graph is complex.
For any graph G the graph G = K 1 + G has r χ (G ) = 1 + r χ (G).

RAINBOW NEIGHBOURHOOD NUMBER OF EDGE-SET GRAPHS
Edge-set graphs were introduced in [4]. As the notion of an edge-set graph seems to be largely unknown. Therefore, the main definition and some important observations from [4] will be presented in this section.
Let A be a non-empty finite set. Let the set of all s-element subsets of A (arranged in some order), where 1 ≤ s ≤ |A|, be denoted by S and the i-th element of S by, A i,s .
is the power set of the edge set E(G). For 1 ≤ s ≤ ε, let S be the collection of all s-element subsets of E(G) and E s,i be the i-th element of S. Then, the edge-set graph corresponding to G, denoted by G G , is the graph with the following properties.
(i) |V (G G )| = 2 ε − 1 so that there exists a one to one correspondence between V (G G ) and E; (ii) Two vertices, say v s,i and v t, j , in G G are adjacent if some elements (edges of G) in E s,i is adjacent to some elements of E t, j in G.
From the above definition, it can be seen that the edge-set graph G G of a given graph G is dependent not only on the number of edges ε, but the structure of G also. Note that it was erroneously remarked in [4] that non-isomorphic graphs of the same size have distinct edge-set graphs. Figure 2 illustrates one contradictory case.
Note that an edge-set graph G G has an odd number of vertices. If G is a trivial graph, then G G is an empty graph (since ε = 0). Also, G P 2 = K 1 and G P 3 = C 3 . In [4] the following conventions were used.
(i) If an edge e j is incident with vertex v k , then we write it as (e j → v k ). such that e i = v i v i+1 , in the sense that v n+1 = v 1 .
Invoking the definition and observations given above, it is noticed that both d t G(e) (G) and The graphs having three edges e 1 , e 2 , e 3 are graphs P 4 ,C 3 , and K 1,3 . The corresponding edge-set graphs Notice that both G C 3 and G K 1,3 are complete graphs.

PROPER COLOURING OF THE EDGE-SET GRAPHS OF PATHS
It is known that for a given size ε ≥ 1 a graph of maximum order ν, is a tree. Hence, for a given size the graphs with maximum structor index si(G) are the corresponding trees, T . It easily follows that for ε(T ) ≥ 3 only the star graphs have G S ε+1 , complete. Put another way, a tree T has G T complete if and only if diam(T ) ≤ 2. From the family of trees, a path corresponding to a given ε, denoted by P ε , has largest diameter. These observations motivate a detailed study of the proper colouring and associated colour parameters of edge-set graphs of paths to lay the foundation for studying more complex graph classes.
For this section paths of the form P n+1 = v 1 e 1 v 2 e 2 v 3 · · · e n v n+1 , will be considered. Such graph will be abbreviated to P n+1 = v 1 e i v i , 1 ≤ i ≤ n. To easily relate the results with Definition 2.1, note that ε(P n+1 ) = n. It can be easily verified that G P 2 = K 1 . Hence, χ(G P 2 ) = 1.
Also, G P 3 = K 3 and hence, χ(G P 3 ) = 3. These observations bring the main results. First, we state an important lemma.
where P(E) is the power set of the edge set E(G). Then each edge e i is in exactly 2 ε−1 subsets of E.
Proof. The result follows directly from the well-definedness and well-ordering of the power set, It is observed that if the number of subsets which has say, e i as element is t, then within the corresponding t subsets the edge e j , j = i will be in t 2 = 2 ε−2 of those subsets.
Theorem 3.2. The edge-set graph G P n+1 , n ≥ 1 has Proof. Part 1: Trivial is the observation that G P 2 = K 1 and that result in equality. It has been observed that G P 3 = K 3 and hence χ(G P 3 ) = 3. Part 3: For n ≥ 4, and the path path P n+1 the edge-set graph G P (n−1)+1 of the preceding path hence, the (n − 1)-edge path P (n−1)+1 , is incomplete. In accordance with the procedure described in Part 2, consider G P (n−1)+1 and G P (n−1)+1 . Since in G P (n−1)+1 the edge e n has been added to each vertex corresponding to the vertices v i, j ∈ V (G P (n−1)+1 ), the new edges in accordance with Definition 2.1 are those between all pairs of vertices for which at least one vertex has e n−1 ∈ v i, j . From Lemma 3.1, it follows that at least one complete induced subgraph, K 2 n−2 exists in G P (n−1)+1 . All pairs of vertices which has both e n−2 , e n−1 ∈ v i, j is an edge in G P (n−1)+1 so least one complete induced subgraph, K 2 n−2 +1 exists in G P (n−1)+1 . Proceeding to vertices for which edge e n−3 ∈ v i, j and so on until the edge e 1 has been accounted for results in G P (n−1)+1 being complete. Hence, χ(G P (n−1)+1 ) = 2 n−1 − 1.
Therefore, by immediate induction, the result follows for all n ≥ 4.

Corollary 3.3.
(a) Each vertex in an edge-set graph G P n+1 , n ≥ 2 belongs to some maximum clique in G P n+1 .
(c) The edge-set graphs G P n+1 , n ≥ 1 are perfect graphs.
Proof. The results are a direct consequence from the proof of Theorem 3.2.
Theorem 3.4. An edge-set graph G P n+1 , n ≥ 1 is a perfect graph.
Proof. For P 1 , P 2 the result is trivial. From Theorem 3.2 and Corollary 3.3(b) we have, n ≥ 2 and hence it follows that ω(G P n+1 ) = 2 n−1 + 2 n−2 − 2 = χ(G P n+1 ). Hence, an edge-set graph is weakly perfect. From Definition 2.1, it follows that an edge-set graph has a unique maximum independent set X. Furthermore, X is a null graph hence, any subgraph thereof is perfect.
Conjecture 1. The edge-set graphs of acyclic graphs are perfect graphs.

CONCLUSION
Research problem: The notion of a chromatic core subgraph of a graph G was introduced in [9]. We recall that, for a graph G its structural size is measured by its structor index denoted and defined as, si(G) = ν(G) + ε(G). We say that the smaller of graphs G and H is the graph satisfying the condition, min{si(G), si(H)}. If si(G) = si(H) the graphs are of equal structural size but not necessarily isomorphic. A straight forward example is the path, P 4 and the star graph, S 3 . Problem 1. For the edge-set graph G P n+1 , n ≥ 4, determine (G P n+1 ).
The research on set-graphs (see [3]) and edge-set graphs naturally leads to new concepts such as vertex degree sequence set-graphs and colour set-graphs and colour-string set-graphs.
Preliminary definitions are provided below.
(1) If the degree sequence of a graph G of order n ≥ 1 is  (i) |G V (G) | = 2 ν − 1 so that there exists a one to one correspondence between V (G V (G) ) and V (G).
(ii) Two vertices, say v s,i and v t, j , in G V (G) are adjacent if some element(s) (specific vertex degree(s) of G) in v s,i is adjacent to some element(s) of v t, j in G.
It follows easily that for a complete graph K n , n ≥ 1 has its corresponding degree sequence set-graph, a complete graph.    (ii) Two vertices, say v s,i and v t, j , in G C {} (G) are adjacent if some element(s) (specific vertex degree(s) of G) in v s,i is adjacent to some element(s) of v t, j in G.
Clearly, for all graphs G with χ(G) = 2 the colour set-graph is K 3 . ).
Problem 4. Research the properties of the colour-string set-graph corresponding to a chromatic colouring of a graph G.