Graph folding and chromatic number

Given a connected graph G , identify two vertices if they have a common neighbor and then reduce the resulting multiple edges to simple edges. Repeat the process until the result is a complete graph. This process is called folding a graph. We show here that any connected graph G which is not complete folds onto the connected graph K p where p = χ ( G ) , the chromatic number of G . Furthermore, the set of all integers p such that G folds onto K p consist of consecutive integers, the smallest of which is χ ( G ) . One particular result of this study is that a sharp upper bound was obtained on the largest complete graph which a graph can be folded onto.


Introduction
Consider the path P 3 of order 3.We may imagine the two edges of the path to be pieces of thin rectangular plates 1 unit long hinged at the common vertex of the two edges of P 3 .An illustration is given in Figure 1.As is usually done, the vertex-set of a graph G is denoted by V (G), and its edge-set is denoted by E(G).In a graph G, the neighbor-set of a vertex x, denoted by N G (x) or simply N (x), is defined as the set of all vertices y adjacent to x. Definition 1.To fold a graph G means to identify two non-adjacent vertices x and y satisfying N (x) ∩ N (y) 6 = ∅ and then reducing multiple edges to simple edges.We say that G folds onto the graph H if H can be obtained from G by applying a sequence (possibly empty) of foldings.
It follows from the definition that if G folds onto G 1 and G 1 folds onto G 2 then G folds onto G 2 .Also, every graph folds onto itself (by the empty sequence of foldings).
Since any two distinct vertices in a complete graph are adjacent, then the complete graph K n folds onto K n only and to no other graph.
By a sequence of foldings any connected graph folds onto some complete graph because each time we fold, we decrease the number of pairs of nonadjacent vertices.
The above definition of folding can be found in [5].It was first defined by the author in 2001.However, this concept was defined even earlier by Cook and Evans in [3] in 1979 from the notion of graph homomorphisms.In their paper, they defined the following.An elementary homomorphism is an identification of two non-adjacent vertices.By the identification of nonadjacent vertices u and v in a graph G, we mean constructing another graph G 0 from G by removing the vertices u and v and all edges incident with u and v from G and adding a vertex w and edges from w to all vertices adjacent to either u or v.A homomorphism for us means a sequence of elementary homomorphisms, and a simple fold is an elementary homomorphism in which the identified vertices have a common neighbor.Kholy and Esawy [4] defined a similar concept in 2005 and they also used the term folding.
As an example, let us fold the Petersen graph shown in Figure 2 onto the complete graph K 5 .Shown in the same figure is the result of identifying 1 and b, and reducing multiple edges to simple edges.We show here that if a connected graph has chromatic number p then it folds onto the complete graph K p .Furthermore, orders of the complete graphs onto which connected graph folds form a set of consecutive integers, the smallest of which is the chromatic number of the graph.

Folding graphs onto complete graphs
An assignment of colors to the vertices of a graph from a set of k colors such that adjacent vertices get different colors is called a proper k-vertex coloring of the graph, or simply, a k-coloring.
More formally, a k-coloring of a graph G is a mapping λ: If G is a graph with connected components G 1 , G 2 , . . ., G c then a sequence of foldings will transform G to a disjoint union of c complete graphs because each G i folds onto some complete graph by applying a sequence of foldings.For this reason, we confine ourselves to folding connected graphs.
Our first theorem gives the effect of folding on the chromatic number of a graph.The result in Theorem 1 was stated without proof as a corollary to an analogous Theorem from [6] in terms of elementary homomorphism.We provide an explicit proof of this result for completeness.

Proof.
Let H be obtained from G by an {a, b}-folding.Let χ(G) = p and χ(H) = q.Consider a p-coloring α of G using p.Define a (p + 1)coloring β of H as follows: If x ∈ V (H) \ {a, b}, let β(x) = α(x).Color the vertex v = {a, b} in H arising from the identification of a and b using the color β(v) = p + 1.Then χ(H) ≤ p + 1.Now consider a coloring γ of H using q colors.Define a q-coloring λ of G as follows: , where v is the vertex obtained by identifying a and b.Then χ(G) ≤ q.Hence, The next theorem states that we can properly choose a folding so that the chromatic number is preserved.

Proof.
Let G be a connected graph which is not complete.If G is a cycle C n , then n ≥ 4. Let 1, 2, . . ., n be the consecutive vertices of C n .By the {1, n − 1}-folding followed by the {n, 2}-folding we get the cycle C n−2 in case n > 4 and K 2 in case n = 4.In both cases, the chromatic number is preserved.
If G is not a cycle, then χ(G) ≤ ∆, the maximum degree of a vertex in G according to Brooks' theorem [1].Let χ(G) = c and let λ: V (G) → {1, 2, . . ., c} be a coloring of G. Denote by S i the set of vertices in G with color i, i = 1, 2, . . ., c. Let x be a vertex in G with deg(x) = ∆.Since ∆ ≥ c, then x must have two neighbor a and b belong to a common set S i .Let G 1 be the graph resulting from the {a, b}-folding.Clearly, G 1 is c-colorable and so χ(G 1 ) ≤ c.By Theorem 1, χ(G 1 ) ≥ χ(G).Therefore, Theorem 3. A connected graph folds onto the complete graph K 2 if and only if it is bipartite.Furthermore, a connected bipartite graph folds only onto K 2 and to no other complete graph.

Proof.
Let G be a connected bipartite graph.Then V (G) can be partitioned into two sets A and B such that every edge of G is of the form xy where x ∈ A and y ∈ B. By Theorem 4, G folds onto Without loss of generality we may assume that both a and b belong to each G i is a bipartite graph.Therefore, G folds onto K 2 and to no other complete graph.
Conversely, let G be a connected graph that folds onto K 2 .By Theorem 1, χ(G) ≤ χ(K 2 ) = 2. Therefore, χ(G) = 1 or 2. If χ(G) = 1, G is the trivial graph and it cannot fold onto K 2 .It follows that χ(G) = 2 and so G is bipartite. 2 Figure 3 illustrates a way of folding the planar grid P 3 × P 3 onto the complete graph K 2 using a sequence of 7 foldings.It is known, and easy to verify, that the chromatic number of the Petersen graph is 3.The Petersen graph folds onto K 3 , as our next result will show.
Theorem 4. If G is a connected graph with chromatic number χ(G) = p, then G folds onto the complete graph K p and to no other smaller complete graph.

Proof.
The theorem is trivially true if G is complete so assume that G is not complete.By Lemma 2, there exist vertices a and b in the graph G with N (a) ∩ N(b) 6 = ∅ such that the graph G 1 obtained from G by the {a, b}-folding has the same chromatic number as G.We repeatedly apply the Lemma until we obtain a complete graph with chromatic number χ(G).Thus the order of the complete graph is χ(G).Theorem 1 states that folding a graph never decreases the chromatic number.Therefore, the smallest order of a complete graph onto which G folds is χ(G). 2 We have seen that the Petersen graph folds onto the complete graph K 5 and onto K 3 as well.Thus, in general, a connected graph folds onto some complete graph that is not unique.
Theorem 5. Let G be a connected graph and let q be the largest order of a complete graph onto which G folds.Then for every integer r with χ(G) ≤ r ≤ q, G folds onto K r .

Proof.
Let The sum of two graphs G and H, denoted by G+H, is formed by taking the disjoint union of G and H and then adding all edges of the form ab where a ∈ V (G) and b ∈ V (H).Theorem 6.Let G and H be connected graphs.Then G + H folds onto K n if and only if for some integers p and q with p + q = n, G folds onto K p and H folds onto K q .
Proof.In the graph G + H, every vertex in G is adjacent to each vertex in H. Therefore folding G + H is accomplished by identifying vertices in G or vertices in H only. 2

Largest complete folding of a graph
We have seen that every connected graph folds onto a set of complete graphs whose orders form a set of consecutive integers.Thus, there is a largest complete folding of a connected graph.The bound given in the theorem is sharp because its value is 5 for the Petersen graph and the Petersen graph folds onto K 5 .

Figure 2 :
Figure 2: Folding the Petersen graph

Theorem 7 . 2 ¢. 2 ¢ 2 !
Let G be a connected graph of order n and size m.If G folds onto K r then χ(G) ≤ r ≤ We already know that r ≥ χ(G) because of Theorem 4. Let G = G 0 → G 1 → • • • → G k = K rbe a folding of G onto some complete graph K r .In each step, the order decreases by 1 and the size (number of edges) decreases by at least 1.Therefore we must have m − k ≥¡ n−k Setting r = n − k, we have m − n + r ≥ ¡ r ≤ m − n + r r(r − 1) ≤ 2(m − n) + 2r r 2 − 3r − 2(m − n) − n) + 9 ¶ºThis completes the proof of the theorem.2