Cubic Knödel Graphs

• Cubic Knödel graphs are 3-regular, bipartite, vertex-transitive graphs defined for even orders (n ≥ 8) with structured connections between two vertex sets.
• They exhibit precisely determined diameters, distance metrics, and total domination numbers, making them central to studies in graph invariants and extremal properties.
• Their construction using index-distance methods and cyclic-sequence analysis bridges explicit combinatorial techniques with algebraic symmetry in graph theory.

A cubic Knödel graph $W_{3,n}$ is a 3-regular, bipartite, vertex-transitive graph of even order $n \geq 8$, arising as a key case within the broader family of Knödel graphs $W_{\Delta, n}$. These structures are notable for combining algebraic regularity, combinatorial symmetry, and extremal properties that make them prototypical objects in the study of domination, diameter, and related invariants in graph theory. Their precise construction and associated structural results furnish a rich testing ground for the analysis of domination-type parameters and distance-related metrics (Mojdeh et al., 2018, Musawi et al., 2020).

1. Definition and Construction

For every even integer $n \geq 8$, the cubic Knödel graph $W_{3,n}$ is defined on vertex set $V = U \cup V$, where

• $U = \{u_j \mid 0 \leq j \leq n/2-1\}$,
• $V = \{v_j \mid 0 \leq j \leq n/2-1\}$.

Edges are defined by adjacency rules that reflect 3-regularity and bipartition:

• Each $u_j \in U$ is adjacent to $v_j$, $n \geq 8$0, and $n \geq 8$1 modulo $n \geq 8$2.
• Each $n \geq 8$3 is adjacent to $n \geq 8$4, $n \geq 8$5, and $n \geq 8$6 modulo $n \geq 8$7.

Alternatively, labeling vertices as $n \geq 8$8 and $n \geq 8$9 for $W_{\Delta, n}$0, edges join $W_{\Delta, n}$1 to $W_{\Delta, n}$2 for $W_{\Delta, n}$3. The resulting graph is bipartite, 3-regular, and vertex-transitive (Mojdeh et al., 2018, Musawi et al., 2020).

2. Structural Properties

Cubic Knödel graphs exhibit strong algebraic and combinatorial features:

• They are bipartite, with both parts $W_{\Delta, n}$4 and $W_{\Delta, n}$5 of size $W_{\Delta, n}$6.
• Each vertex has degree 3.
• The graph is vertex-transitive.
• For any two vertices in the same part, the intersection of their neighborhoods is at most one, and exactly one if and only if their index-distance is in $W_{\Delta, n}$7.
• $W_{\Delta, n}$8 is $W_{\Delta, n}$9-free (does not contain a complete bipartite subgraph $n \geq 8$0 as an induced subgraph).

Table: Basic Parameters of $n \geq 8$1 | Parameter | Value | Reference | |----------------------------------|----------------------|-------------------| | Order ($n \geq 8$2) | even $n \geq 8$3 | (Mojdeh et al., 2018) | | Degree | 3 | (Mojdeh et al., 2018) | | Vertex-transitive | Yes | (Mojdeh et al., 2018) | | Bipartite | Yes | (Mojdeh et al., 2018) |

3. Distance and Diameter

For $n \geq 8$4, the diameter of $n \geq 8$5 is given by

$n \geq 8$6

This exact bound follows from a general distance analysis of Knödel graphs. The formulas and procedures for computing distances within and between partite sets are as follows (Musawi et al., 2020):

• For $n \geq 8$7:

$n \geq 8$8

where $n \geq 8$9.

• For $W_{3,n}$0, $W_{3,n}$1:

$W_{3,n}$2

• For $W_{3,n}$3, the formula is analogous.

The graph strictly increases in diameter as $W_{3,n}$4 increases by increments of 6. For example, $W_{3,n}$5 all have diameter 3, while $W_{3,n}$6 all have diameter 4 (Musawi et al., 2020).

4. Total Domination Number

A total dominating set $W_{3,n}$7 in $W_{3,n}$8 is a subset such that every vertex has a neighbor in $W_{3,n}$9. The minimum cardinality of such a set, denoted $V = U \cup V$0, is determined exactly for all even $V = U \cup V$1. Writing $V = U \cup V$2,

$V = U \cup V$3

Equivalently,

$V = U \cup V$4

where $V = U \cup V$5 when $V = U \cup V$6 and $V = U \cup V$7 when $V = U \cup V$8 (Mojdeh et al., 2018).

The construction of extremal total dominating sets is explicit: for $V = U \cup V$9, select $U = \{u_j \mid 0 \leq j \leq n/2-1\}$0, and for certain congruence classes of $U = \{u_j \mid 0 \leq j \leq n/2-1\}$1 additional vertices are chosen to cover remaining uncovered vertices.

This result, completing the determination of total domination in the cubic case (the ordinary domination number $U = \{u_j \mid 0 \leq j \leq n/2-1\}$2 had been previously obtained for all $U = \{u_j \mid 0 \leq j \leq n/2-1\}$3), serves as a paradigm for domination-type invariant analyses in regular bipartite graphs.

5. Proof Techniques and Combinatorial Tools

The bounds on $U = \{u_j \mid 0 \leq j \leq n/2-1\}$4 utilize both explicit constructions and lower bounds established via combinatorial arguments:

• The pigeonhole principle partitions any candidate set into $U = \{u_j \mid 0 \leq j \leq n/2-1\}$5 and $U = \{u_j \mid 0 \leq j \leq n/2-1\}$6, observing that each $U = \{u_j \mid 0 \leq j \leq n/2-1\}$7 has three neighbors in $U = \{u_j \mid 0 \leq j \leq n/2-1\}$8.
• Sharpness in counting arguments is achieved by introducing the cyclic-sequence of subset $U = \{u_j \mid 0 \leq j \leq n/2-1\}$9 and the index-distance function $V = \{v_j \mid 0 \leq j \leq n/2-1\}$0.
• Lemma 2.3 specifies necessary and sufficient conditions for vertices in $V = \{v_j \mid 0 \leq j \leq n/2-1\}$1 to have common neighbors in $V = \{v_j \mid 0 \leq j \leq n/2-1\}$2, in terms of membership of the index-distance in $V = \{v_j \mid 0 \leq j \leq n/2-1\}$3.
• Lemmata 2.7 and 2.8 detail neighborhood intersections and control the gaps of cyclic-sequences, making extremal arguments on dominating sets tight.

These methods are not only instrumental for the cubic ($V = \{v_j \mid 0 \leq j \leq n/2-1\}$4) case but are suggested to generalize toward higher-regularity Knödel graphs $V = \{v_j \mid 0 \leq j \leq n/2-1\}$5 for $V = \{v_j \mid 0 \leq j \leq n/2-1\}$6 (Mojdeh et al., 2018).

6. Examples and Special Cases

Specific computations for small values of $V = \{v_j \mid 0 \leq j \leq n/2-1\}$7 illustrate the structural results:

• For $V = \{v_j \mid 0 \leq j \leq n/2-1\}$8, $V = \{v_j \mid 0 \leq j \leq n/2-1\}$9 (direct verification).
• For $u_j \in U$0, $u_j \in U$1, so $u_j \in U$2 (example: $u_j \in U$3).
• For $u_j \in U$4, $u_j \in U$5, $u_j \in U$6, so $u_j \in U$7 (example: $u_j \in U$8).

There is a unique threshold value for the diameter: for $u_j \in U$9, $v_j$0 is exceptional, with diameter 3, divergent from the formula valid for $v_j$1 (Musawi et al., 2020).

7. Broader Implications and Related Directions

Cubic Knödel graphs constitute canonical instances of highly regular, symmetric, bipartite graphs amenable to explicit combinatorial analysis. The mutual constraints between local and global structure (e.g., $v_j$2-freeness, sharp dominating sets, explicit diameter) make them central objects for exploring extremal questions in graph domination and communication. The machinery developed extends naturally to domination-type invariants and suggests that combinatorial techniques—such as cyclic-sequence analysis and index-distance tools—are broadly applicable to deeper studies of $v_j$3-regular bipartite graphs and their algorithmic properties (Mojdeh et al., 2018, Musawi et al., 2020).