Light subgraphs in graphs with average degree at most four

A graph H is said to be light in a family G of graphs if at least one member of G contains a copy of H and there exists an integer λ(H,G) such that each member G of G with a copy of H also has a copy K of H such that degG(v) ≤ λ(H,G) for all v ∈ V(K). In this paper, we study the light graphs in the class of graphs with small average degree, including the plane graphs with some restrictions on girth. We proved that: 1. If G is a plane graph with δ(G) ≥ 2 and face size at least 7, then G has a (2, 2, 5−)-, (2, 5−, 2)or (3, 3, 2, 3)-path. 2. If G is a graph with δ(G) = 2, average degree less than 3 and without (2, 2,∞)-triangles, then G has one of the following configurations: a (2, 3−, 3−)-path, a (2, 2,∞, 2)-path, a (2, 2, 4, 3)-path, a (4; 2, 2, 2, 6−)-star, a (4; 2, 2, 3, 5−)-star, a (4; 2, 3, 3, 3)-star, a (5; 2, 2, 2, 2, 2)-star, and a (5; 2, 2, 2, 2, 3)-star. 3. If G is a graph with δ(G) = 2, average degree less than 3 and without (2, 2,∞)-triangles, then G has one of the following configurations: a (2, 3−, 3−)-path, a (2, 2,∞, 2)-path, a (2, 2, 4, 3)-path, a (4; 2, 2, 2, 6−)-star, a (2, 4, 3, 2)-path, a (2, 4, 3)-triangle, a (5; 2, 2, 2, 2, 2)-star, and a (5; 2, 2, 2, 2, 3)-star. 4. If G is a graph with δ(G) ≥ 2 and average degree less than 10 3 , then G has one of the following configurations: a (2, 2,∞)-path, a (2, 3, 6−)-path, a (3, 3, 3)-path, a (2, 4, 3−)-path, and a (2, 9−, 2)-path. 5. If G is a graph with δ(G) = 3 and average degree less than 4, then G contains a (4−, 3, 7−)-, (5, 3, 5)or (5, 3, 6)path. 6. If G is a triangle-free normal plane map, then it contains one of the following configurations: a (3, 3, 3)-path, a (3, 3, 4)-path, a (3, 3, 5, 3)-path, a (4, 3, 4)-path, a (4, 3, 5)-path, a (5, 3, 5)-path, a (5, 3, 6)-path, and a (3, 4, 3)path.