Cycle Double Covers in Cubic Graphs having Special Structures

Abstract

In the first part of this article, we employ Thomason's Lollipop Lemma to prove that bridgeless cubic graphs containing a spanning lollipop admit a cycle double cover (CDC) containing the circuit in the lollipop; this implies, in particular, that bridgeless cubic graphs with a 2‐factor F having two components admit CDCs containing any of the components in the 2‐factor, although it need not have a CDC containing all of F. As another example consider a cubic bridgeless graph containing a 2‐factor with three components, all induced circuits. In this case, two of the components may separately be used to start a CDC although it is uncertain whether the third component may be part of some CDC. Numerous other corollaries shall be given as well. In the second part of the article, we consider special types of bridgeless cubic graphs for which a prominent circuit can be shown to be included in a CDC. The interest here is the proof technique and therefore we only give the simplest case of the theorem. Notably, we show that a cubic graph that consists of an induced 2k‐circuit C together with an induced 4k‐circuit T and an independent set of 2k vertices, each joined by one edge to C and two edges to T, has a CDC starting with T.

Cycle double covers containing certain circuits in cubic graphs having special structures

“…We observe that the graphs in (i) have been treated in [2], whereas some cubic graphs in (ii), (iii) are shown in the present paper to have CDCs. Note that in the cubic case, the graphs of (ii) considered in Theorem 1 can be viewed as generalizations of the graphs of (i).…”

“…The Cycle Double Cover Conjecture (CDCC) has found a lot of attention since the 1970s (see, e.g., the introduction of [2]), and there is even a book on Cycle Double Covers (CDCs) by C.-Q. Zhang [3].…”

“…Our next aim is to use (2) (i.e., the existence of the T -join just proved), to produce a cover of G by two subgraphs G * r and G * b which are homeomorphic to respective 3-edge-colorable cubic graphs H r and H b . The latter will have respective 2-factors Q r and Q b whose circuits correspond to C 1 and to vertices in H. Now, c corresponds in G to the circuit C which is connected to C 1 by an even number of edges due to the congruence (2). Observe that C corresponds to a component in G r,y or to a component in G b,y .…”

“…(1) T is an induced caterpillar, C ′ and C ′′ are induced circuits; (2) x and y are the ends of a spine of T ;…”

Cycle double covers via Kotzig graphs

“…Other spanning subgraphs than Kotzig graphs and even cycles also imply the existence of a CDC. For more information regarding this approach towards a solution of the CDCC, we refer to [2,4,6]. Note that not every 3-connected cubic graph has a Kotzig-frame, see [5].…”