On Spanning and Dominating Circuits in Graphs

Abstract

A set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

Hamilton cycles and closed trails in iterated line graphs

“…Other papers on Hamiltonian cycles in iterated line graphs, or in sub-5 graphs of iterated line graphs, include [1], [2], [7], [9], [10], [11], and [12].…”

“…, where e is incident with a vertex of degree 1 in G. Then |E(B j )|, (12) and such that the subgraph H defined by…”

Supereulerian graphs: A survey

“…Numerous sufficient conditions for G ∈ SL have been expressed in terms of lower bounds on degrees in G (see, e.g., [12], [26], [27], [28], [31], [32], [34], [36], [53], [56], [57], [58], [59], [104], [105], [106], [120], [144], [148]). Some of these sufficient conditions even imply that G satisfies the hypothesis of Theorem 4.1, and the hypothesis of that theorem can be satisfied by graphs with far fewer edges.…”

“…For other sufficient conditions for a graph to have a dominating trail, see [5], [6], [7], [9], [10], [11], [12], [13], [14], [15], [23], [24], [25], [26], [45], [46], [53], [54], [49], [51], [59], [61], [62], [82], [90], [92], [96], [97], [98], [100], [103], [104], [108], [113], [114], [115], [119], [143], [144], [145], [146], [148], [149]. The reduction method is not too useful for finding dominating cycles or paths, but it can be applied to find dominating trails.…”

“…The subdivision graph S 1 (G) is obtained from G by inserting a new vertex into each edge. Lesniak-Foster and Williamson [104] and S.-M. Zhan [149] noted that a graph G has a dominating open trail if and only if L(G) has a hamilton path. Theorem 2.3 (Jaeger [85]) Let G be a graph and let E ⊆ E(G).…”