Let H = (V, E) be a hypergraph with vertex set V and edge set E of order n H = |V | and size m H = |E|. A transversal in H is a subset of vertices in H that has a nonempty intersection with every edge of H. A vertex hits an edge if it belongs to that edge. The transversal game played on H involves of two players, Edge-hitter and Staller, who take turns choosing a vertex from H. Each vertex chosen must hit at least one edge not hit by the vertices previously chosen. The game ends when the set of vertices chosen becomes a transversal in H. Edge-hitter wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game transversal number, τ g (H), of H is the number of vertices chosen when Edge-hitter starts the game and both players play optimally. We compare the game transversal number of a hypergraph with its transversal number, and also present an important fact concerning the monotonicity of τ g , that we call the Transversal Continuation Principle. It is known that if H is a hypergraph with all edges of size at least 2, and H is not a 4-cycle, then τ g (H) ≤ 4 11 (n H + m H ); and if H is a (loopless) graph, then τ g (H) ≤ 1 3 (n H + m H + 1). We prove that if H is a 3-uniform hypergraph, then τ g (H) ≤ 5 16 (n H + m H ), and if H is 4-uniform, then τ g (H) ≤ 71 252 (n H + m H ).
The triangle graph of a graph G, denoted by T (G), is the graph whose vertices represent the triangles (K 3 subgraphs) of G, and two vertices of T (G) are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a characterization of C n -free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graphs G for which T (G) is a tree, a chordal graph, or a perfect graph. For the class of graphs whose triangle graph is perfect, we verify a conjecture of the third author concerning packing and covering of triangles.
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