LINE TRIANGLE INTERSECTION GRAPH

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. A graph operator is a mapping between two families of graphs. In this paper, a new graph operator called the line triangle intersection graph is introduced. Also, the concept of a quasi regular graph is proposed. Further, various properties of line triangle intersection graph of a graph are investigated including its chromatic number and clique number. It is proved that the line triangle intersection graph of a complete graph is quasi regular. Moreover, partial characterization for a line triangle intersection graph is presented.


INTRODUCTION
Even though Whitney [3] used the construction of line graphs, it was Krausz [4] who formulated the concept of a graph operator and that of a line graph. A characterization of line graphs was also given by him. Later, Beineke [5] gave a new characterization of line graph in terms of 9 forbidden subgraphs. AgainŠoltés [6] gave forbidden induced subgraphs of a line graph with atleast 9 vertices. Modifying the construction of a line graph, two new graph Definition 2.17. [7] The Gallai graph Γ of a graph G is the graph whose vertex-set is the edgeset of G; two distinct edges of G are adjacent in Γ(G) if they are incident in G, but do not span a triangle in G.
Definition 2.18. [7] The anti-Gallai graph of a graph G denoted as ∆(G) has the edges of G as its vertices; two edges are adjacent in ∆(G) if they span a triangle in G.
(Throughout this paper ∆(G) represent maximum degree of a vertex in G) Theorem 2.19. [7] For every graph G with ω(G) ≥ 1, where γ(G) denote anti-Gallai graph of G Theorem 2.20. [5] The following statements are equivalent for a graph G.
(1) G is the derived graph of some graph.
(2) The edges of G can be partitioned into complete subgraphs in such a way that no vertex belongs to more than two of the subgraphs.
(3) The graph K 1,3 is not an induced subgraph of G;and if abc and bcd are distinct odd triangles, then a and d are adjacent.
Theorem 2.21. Brook's Theorem [8]: Every graph G with maximum degree ∆(G) has a ∆(G) coloring unless either (i) G contains K ∆(G)+1 or (ii) ∆(G) = 2 and G contains an odd cycle. A vertex corresponding to an edge in G is called an edge vertex and a vertex corresponding to a triangle in G is called a triangle vertex.  Definition 3.3. A graph having only two distinct non-negative integers in its degree sequence is called a quasi-regular graph.

LINE TRIANGLE INTERSECTION GRAPH
Example 3.4. In Figure 3, the degree of red vertices is 3 and the degree of green vertices is 2.
So, it is a quasi-regular graph.  incident on an edge is 0, the T-degree of that edge is considered to be zero. The T-degree of an edge 'e' is denoted as T d (e).
Example 3.7. In figure 4, the T-index of the graph is 3. Also, Proof. By definition itself, LT (K 1,n ) = K n .
Now if LT (G) = K n for some G, then G must be triangle free. Also each edge in G should be adjacent to all other edge. This is possible only when G = K 1,n .
Theorem 3.14. LT (G) is isomorphic to G iff G = n>3 C n or G = C 3 ∪ K 1 Proof. Necessary: If G is a triangle free graph, then LT (G) = L(G) and we know that L(G) is isomorphic to G iff G = C n . So, let us assume that G is a connected graph having triangles. Now, suppose that LT (G)isomorphic to L(G). This is possible only if n = m +t where m denote number of edges of G and t denotes T-index of G. Now consider the least case when t = 1, ie; there is only one triangle . Then, n = m + 1 or m = n − 1. ie; G is a tree which is a contradiction.
When t>1, n<m − 1 which means that G is disconnected. Again a contradiction. So, LT (G) is not isomorphic to G except when G = C 3 ∪ K 1 Sufficient: C n is a triangle free graph and Proof. If G 1 and G 2 are isomorphic, there is a one-one correspondence between the vertex set of G 1 and G 2 and a one one-one correspondence between the edge set of G 1 and G 2 which preserves the incidence and adjacency relation. So, there will be a one-one correspondence between the triangles of G 1 and G 2 . Hence, in the line-triangle intersection graph of G 1 and G 2 also there will be a one-one correspondence between the vertex sets and the edge sets which preserves adjacency and incidence relation. Hence, LT (G 1 ) is isomorphic to LT (G 2 ) Theorem 3.17. For any graph G with ω(G) ≥ 1, Proof. Let ω(G) = 3, 4. If K n is a complete subgraph of G where n = 3, 4, then 'n − 1' edges of K n have a common endpoint in G. These 'n − 1' edges form a complete graph in LT (G).
From theorem 2.19 ∆(G) = ω(LT (G)). ∴ ∆(G) ≤ χ(LT (G)) Theorem 3.20. For any graph G, Proof. Let P be the set of edge vertices and T be the set of triangle vertices of LT (G).
Here, there are two cases.
Case 1: G is a triangle free graph.
Then, the vertex set of LT (G) consists of P only. Hence, each vertex p in LT (G) corresponds to an edge uv in G.
Then, the degree of t in LT (G), According to Brook's theorem, χ(LT (G)) ≤ ∆(LT (G)) + 1, and hence Theorem 3.21. Let G be a graph with n vertices. Then, LT (G) can be partitioned into edge disjoint complete graphs in such a way that no vertex is common to more than (n − 2) 2 of such subgraphs.
Proof. Case 1: Let t be a triangle vertex. Then, vertices in LT (G), corresponding to the edges incident at each vertex of the triangle corresponding to t in G, induces a complete subgraph along with t in LT (G). Since t has exactly 3 vertices, t is common to atmost 3 complete subgraphs in LT (G), for n ≥ 6 Case 2(a): Let e be an edge in G which is not incident with any triangle in G. Then, the edges incident at each end vertex of 'e' induces a complete subgraph in LT (G). So, 'e' will be common to maximum of two complete subgraphs of LT (G).
Case 2(b): Let 'e' be an edge in G which is incident with one or more triangles in G. Then, each vertex corresponding to a triangle incident with e, alongwith the vertices corresponding to e and the other edges incident on the common vertex of e and the triangle, induce a complete subgraph in LT (G). Now, maximum number of triangles incident on each end vertex of 'e' will be (n−1) C 2 . Of these, (n−1) C 2 + (n−1) C 2 triangles, (n−2) C 1 triangles will be common. Therfore, the maximum number of triangles incident on an edge 'e' in a graph will be (n−1) C 2 + (n−1) C 2 −