Equivalence classes, solitary patterns and cubic graphs

• Main Authors: Narayana, D. V. V, Gohokar, Kalyani, Kothari, Nishad
• Format: Journal Article
• Language: English
• Published: 31-08-2024
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2409.00534

A nontrivial connected graph is matching covered if each edge belongs to some
perfect matching. There is extensive theory on these graphs; see Lucchesi and
Murty [Perfect Matchings, Springer 2024]. Two edges are mutually dependent if
every perfect matching either contains both or neither. This is an equivalence
relation and induces a partition of the edges; each part is called an
equivalence class. Note that $\frac{n}{2}$ is an upper bound on the size of any
equivalence class. We characterize those graphs that attain this upper bound,
thus fixing an error in [Discrete Mathematics, 343(8), 2020].
Sch\"onberger (1934) strengthened the famous result of Petersen by proving
that every 2-connected cubic graph is matching covered. This immediately begs
the question as to which 2-connected cubic graphs have the property that each
edge belongs to two or more perfect matchings. An edge is solitary if it
belongs to exactly one perfect matching. Note that if any member of an
equivalence class is solitary then so is each member; such an equivalence class
is called a solitary class. This leads us to the solitary pattern -- the
sequence of sizes of solitary classes in nonincreasing order. For instance,
$K_4$ has solitary pattern (2,2,2) whereas the Petersen graph has solitary
pattern ().
Using the theory of Lucchesi and Murty, we prove that every 3-connected cubic
graph $G$ has one of the following solitary patterns: (2,2,2), (2,2,1), (2,2),
(2,1,1), (2,1), (2), (1,1,1), (1,1), (1) or (); consequently, $G$ has at most
six solitary edges. We also provide characterizations of 3-connected cubic
graphs that have a given solitary pattern except for (1,1), (1) and ().
Finally, we prove that every 2-connected cubic graph, except $\theta, K_4,
\overline{C_6}$ and the bicorn, has at most $\frac{n}{2}$ solitary edges, and
equality holds if and only if its solitary edges comprise a perfect matching.