The total graph of a hypergraph

Let H be a hypergraph with vertices V(H) and hyperedges E(H). The total graph of H, T(H), is the simple graph with vertex set V(H)U E(H) where vertices x and y of T(H) are adjacent if and only if x is contained in, contains or is adjacent to y in H. We give a simple characterisation of those graphs which are the total graphs of some hypergraphs. We show that the total graph uniquely defines a linear hypergraph up to isomorphism and duality and present examples to show that this is not the case for general nonlinear hypergraphs. We give a polynomial time algorithm for the problem of deciding whether a given graph is the total graph of a linear hypergraph. 1. I n t r o d u c t i o n A hypergraph H is a pair (V (H) ,E (H) ) , where V(H) is a finite set of vertices and E(H) is a finite family of nonempty subsets of V(H) called hyperedoes or just edges, with UE~E(H)E = V(H). H is linear if for all distinct E, E ' E E(H), IE N E'[ ~< 1, so for a linear hypergraph there may be no repeated hyperedges of cardinality greater than one. Distinct vertices v, v' E V(H) are adjacent if there is some hyperedge E E E(H) with v, v / E E. Distinct hyperedges E, E ' E E(H) are adjacent if E N E' ¢ (~. Vertex v E V(H) is incident with hyperedge E E E(H), and vice versa, if v E E. A path from v C V(H) to v' E V(H) is a finite sequence (v, EI,w1,E2, w 2 . . . . . Ek_l,Wk_.l,Ek,v I) such that v E El, wi E Ei NEi+I for i = 1,2 . . . . . k 1 and v ~ E Ek. H is connected if for every pair v, v' E V(H) there is some path from v to v'. The dual of H = ({vl,v2 . . . . . vn}, [EI,E2 . . . . . Em]), H*, is the hypergraph whose vertices {el,e2 . . . . . em} correspond to the hyperedges of H, and with hyperedges V i = { e / : v i E E j i n H } ( i = 1,2 . . . . . n). 1 Research supported by the SERC. * E-mail: [email protected]. 0012-365X/97/$17.00 Copyright (~) 1997 Published by Elsevier Science B.V. All rights reserved PH S00 12-3 65X(96)00230-0 216 P. Cowling~Discrete Mathematics 167/168 (1997) 215-236 The rank of H, rank(H), is the maximum cardinality of a hyperedge in E(H). A hyperedge of rank one is a loop. The degree of a vertex v E V(H), deg/4(v), is the number of hyperedges containing v. The maximum degree among vertices of H is denoted A(H). A simple 9raph is a linear hypergraph of rank 2 without loops. A multigraph with loops is a hypergraph of rank 2. Note that under our definition these graphs may not have isolated vertices. When talking of the edge {v,w} of a graph, we will often write simply vw. The subgraph of graph G = (V(G),E(G)) induced by W C_ V(G) is the graph with vertex set W containing only those edges of E E E(G) with E C_ W. The distance from v E V(G) to w E V(G), dc(v,w) is the minimum number of edges in a path from v to w. The neighbour set of a vertex v E V(G), NG(v), is the set of vertices at distance 1 from v. The closed neighbour set of v, NG(v), is No(v) U {v}. A bipartite graph is a simple graph G whose vertex set V(G) has a bipartition (S, T) such that S and T both induce a graph with no edges. The complete 9raph on n vertices, Kn is a simple graph on n vertices where every pair of vertices is adjacent. The cycle on n vertices Cn is a connected simple graph on n vertices where every vertex has degree 2. Two graphs Gi, G2 are isomorphic (written Gi ~ G2) if there is some bijection 0 : V(Gl) ~ V(G2) such that vw E E(G1) ¢~ O(v)O(w) E E(G2). Note Cn ~C,~. The cycles are the only family of graphs for which this is the case. Note that the dual G* of a graph G is not a graph unless G has maximum degree 2. Associated with the hypergraph H we have several graphs. The 2-section of H, He, is the simple graph with vertex set V(H) where distinct x,y E V(H) are adjacent in H2 if and only if they are adjacent in H. The line 9raph of H, L(H), is the simple graph with vertex set E(H) where distinct E,U E E(H) are adjacent in L(H) if and only if E and E' are adjacent in H. Then L(H) ~ (H*)2. The incidence 9raph of H, I (H) , is the bipartite graph with vertices V(H)U E(H) and bipartition (V(H),E(H)) where v E V(H) is adjacent to E E E(H) if and only if v is contained in the hyperedge E of H. Then I(H) ~I(H*) and I(H) uniquely defines H up to isomorphism and duality. Two hypergraphs HI, H2 are isomorphic (written HI ~ /42) if I(H1 ) ~ I(H2) and the isomorphism 0 maps V(H1 ) onto V(H2) and maps E(HI ) onto E(H2). H1 is dual isomorphic to H2 if H1 ~ H~*. The total 9raph of H, T(H), is the simple graph with vertices V(H)U E(H) where x, y E (V(H) U E(H) ) are adjacent if and only if x is contained in, contains or is adjacent to y in H. The edge set of T(H) is the disjoint union of the edge sets of H2, L(H) and I(H) and we have thus T(H)~ T(H*). The middle 9raph o f / / , M(H) (see [5,13]) is the subgraph of T(H) formed by deleting all edges connecting pairs of vertices of V(H). A (strony) vertex colourin9 of hypergraph H is a mapping C : V(H) ~ {1,2 . . . . . k} such that every pair of adjacent vertices receives different colours. The smallest k for which a vertex colouring exists is the chromatic number z(H). For all hypergraphs H we have that z (H) = z(H2). A total colourin9 of H is a mapping C : (V(H)UE(H)) --~ { 1,2 . . . . . k T } such that every pair of adjacent vertices, every pair of adjacent hyperedges P. Cowling / Discrete Mathematics 167/168 (1997) 215-236 217 and every incident vertex and hyperedge receive different colours. The smallest k~ tbr which such a colouring exists is the total chromatic number zT(H). Note that a total colouring of H defines a total colouring of H*, hence zT(H) = z:(H*). This 'selfduality' is one of the most useful properties of total colourings of hypergraphs, which we will use repeatedly in this paper. The total graph of a hypergraph arises since for all hypergraphs H, z r (H) = z(T(H)) . The study of the total chromatic number for hypergraphs and in particular linear hypergraphs, is motivated in part by the total colouring conjecture, posed independently by Behzad [1] and Vizing [14], which we now give. Total colouring conjecture (Behzad [1] and Vizing [14]). Let G be a simple graph. Then