The Transversal Of The Odd Cycles Hypergraphs And Even Edge Coloring Of A Hypergraph

Hypergraph is a subset system of finite sets. The theory of hypergraphs is growing rapidly. The coloring theory of hypergraph plays an important role in discretely-divisible mathematics. It is well know that the trees of graph are applied widely in computer fields. By way of the generalization of graph, the cycles of hypergraph were investigated early. In 1965 Lovaze prove that if a hypergraph H of order n and with p connected components contains no cycles of length≥3 and no loops and if |E_i∩E_j|≤2 for E_i≠E_j then(?)(|E_i|-2)2 is a useful concept to study the parity problems of G. In this section, the concept is spread into hypergraph, form the definition of H+K₂. Theorem 2.2.1 obtained thatTheorem 2.2.1 There exists in H an odd chain from the vertex x to the vertex z if the hypergraph obtained from H + K₂ by removing the vertices x' and z' has a perfect matching.Theorem 2.2.2 presented thatTheorem 2.2.2 The transversals of the odd cycles of hyper graphs H with hereditary 2-coloringÏ„'(H)=n-α(H+K₂).There has been a lot of conclusions about stability numberαwhich had been known. Thus the calculation ofÏ„'(H) was predigested. Then we were interested in the hypergraphs with hereditary 2-coloring. In the subsection 2.3, we main discussed the hypergraphs with hereditary 2-coloring. Theorem 2.3.1 proved thatTheorem 2.3.1 TheÏ„-uniform hypergraph with hereditary 2-coloring if all coefficient of the coloring polynomials q(H) is not equal to zero mod k. Theorem 2.3.2 proved thatTheorem 2.3.2 If the coloring polynomials q(H) of linear hypergraph H contains monomial x_k,(k∈{1, 2,···,n}), then H with hereditary 2-coloring.In the end, we defined the hypergraph Z_H, and presented theorem 2.3.4,Theorem 2.3.4 If the hypergraph H is normal, then the chromatic number of H is≤4.In section 3: The concept of the even edge coloring of a graph was motivated from a study on balanced signed graphs. In this section, the concept is spread into hypergraph, form the definition of the even edge coloring of a hypergraph, and main studied the maximum even edge coloring of a hypergraphε(H). Theorem 3.2.1 proved thatTheorem 3.2.1 If H is connectedÏ„-uniform hypergraph of order n, then the upper bound of the maximum even edge coloring of H is (?).And it is easy to get thatCorollary 3.2.1 If H isÏ„-uniform hypergraph of order n with k components, then the upper bound of the maximum even edge coloring of H is (?).In the end, we proved thatTheorem 3.2.3 If H isÏ„-uniform hypergraph of order n and contains only cycle, thenε_max(H)=(?)-1. If G is a graph of order n with k components, thenε_max(G)=n-k. The result of theorem 3.2.1 and 3.2.3 is identical whenÏ„=2. This accords with the conclusion in graph. And whenÏ„>2, The result of theorem 3.2.1 and 3.2.3 showed the intuitive conjecture, i.e the maximum even edge coloring ofÏ„-niform hypergraph H of order n with k components is (?),uncorrect.