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 lambda(H, G) such that each member G of G with a copy of H also has a copy K of H such that deg(G)(v) <= lambda(H, G) for all v is an element of 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 graph with delta(G) = 2, average degree less than 14/5 and without (2, 2, infinity)-triangles, then G has one of the following configurations: a (2, 2, 13(-), 2)-path, a (2, 3(-), 3(-))-path and a (4; 2, 2, 2, 3(-))-star. 2. If G is a plane graph with delta(G) >= 2 and face size at least 7, then G has a (2, 2, 5(-))-path, or a (2, 5(-), 2)-path or a (3, 3, 2, 3)-path. 3. If G is a graph with delta(G) = 2, average degree less than 3 and without (2, 2, infinity)-triangles, then G has one of the following configurations: a (2, 3(-), 3(-))-path, a (2, 2, infinity, 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. 4. If G is a graph with delta(G) = 2, average degree less than 3 and without (2, 2, infinity)-triangles, then G has one of the following configurations: a (2, 3(-), 3(-))-path, a (2, 2, infinity, 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. 5. If G is a graph with delta(G) >= 2 and average degree less than 10/3, then G has one of the following configurations: a (2, 2, infinity)-path, a (2, 3, 6(-))-path, a (3, 3, 3)-path, a (2, 4, 3(-))-path and a (2, 9(-), 2)-path. 6. If G is a graph with delta(G) = 3 and average degree less than 4, then G contains a (4(-), 3, 7(-))-path, or a (5, 3, 5)-path or a (5, 3, 6)-path. 7. 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. (C) 2016 Elsevier B.V. All rights reserved.
