Counting Circuit Double Covers

Abstract

We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to $C_k$ for some $k$) instead of cycles (graphs with all degrees even). We give an almost-exponential lower-bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower-bound for planar graphs. We conjecture that any bridgeless cubic graph has at least $2^{n/2-1}$ circuit double covers and we show an infinite class of graphs for which this bound is tight.

Figures (11)

• Figure 1: CiDC around a vertex of degree 3
• Figure 2: Two possible CiDCs of a triangle gadget
• Figure 3: A drawing of $K_4$
• Figure 4: The basic idea of the flower construction. Circles denote the possible choices.
• Figure 5: Flower 4-pole with its two outer-fixed CiDCs we consider.

Theorems & Definitions (40)

• Theorem 1: Esperet et al. EKKKN-expmatch
• Theorem 2: Thomassen Thom-exp5-col
• Theorem 3: Dvořák, Mohar and Šámal DMS19
• Conjecture 4: Szekeres '73 sze73, Seymour '79 sey79
• Conjecture 5
• Definition 6: Circuit and Cycle
• Definition 7: Double Cover
• proof
• proof
• Definition 10: Cyclic Connectivity