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Weak immersion is a generalization of the immersion relation defined by Nash-Williams. A graph H is said to be weakly immersed in a graph G if H can be obtained from G by a sequence of these three operations: taking a subgraph, splitting a vertex, and lifting a pair of adjacent edges. The weak...
Cai an Corneil (Discrete Math. 102 (1992) 103–106), proved that if a graph has a cycle double cover, then its line graph also has a cycle double cover, and that the validity of the cycle double cover conjecture on line graphs would imply the truth of the conjecture in general. In this note we...
Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the graph minor theorem . In this paper we characterize several families of graphs with small feedback sets, namely k 1 -F EEDBACK V ERTEX S ET , k 2 -F eedback E DGE S ET and ( k 1 , k 2 )-F...
The object of this paper is to review the general problem of using degree conditions to determine the structure of 2-factors in graphs. We shall discuss open problems and developments in this area and to a very limited extent, provide examples of the proof techniques used. We shall also consider...
In previous papers, Catlin introduced four functions, denoted S O , S R , S C , and S H , between sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k -flows for a...
For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K 1,4 -free graph G with minimum degree δ ( G )⩾ k +3 is pancyclic mod k and every claw-free graph G with δ ( G )⩾ k +1 is pancyclic mod k ,...
For an integer l ⩾0, define SE (l) to be the family of graphs such that G∈ SE (l) if and only if for any edge subset X ⊆ E ( G ) with | X |⩽ l , G has a spanning eulerian subgraph H with X ⊆ E ( H ). The graphs in SE (0) are known as supereulerian graphs. Let f ( l ) be the minimum...
In this survey type article, various connections between eulerian graphs and other graph properties such as being hamiltonian, nowhere-zero flows, the cycle-plus-triangles problem and problems derived from it, are demonstrated. It is also shown how compatible cycle decompositions can be used to...
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