Combinatorial Prisms in Graph Theory

• Combinatorial Prisms are two-layered constructions formed by taking the Cartesian product of a graph or simplex with a two-element structure, unifying diverse combinatorial concepts.
• They facilitate studies in Hamiltonicity, threshold colorings, automorphism groups, and yield combinatorial bijections across geometric and algebraic frameworks.
• Applications span Bubble-sort graph modules, simplicial triangulations, Ehrhart theory in prism slices, and the enumeration of inscribed polyominoes.

Combinatorial prisms are product-type or matching-type constructions that create a two-layered combinatorial object from a graph, simplex, or rectangular box. In the literature surveyed here, the term includes generalized prisms $H\Box K_2$ inside Bubble-sort graphs, ordinary prism graphs $C_n\times P_2$, complementary prisms $\Gamma\overline{\Gamma}$, simplicial prisms $\Delta_n\times\Delta_1$, and slices or inscribed substructures of rectangular prisms. Across these settings, prisms serve as organizing devices for Hamiltonicity, threshold colorings, automorphism groups, representation-theoretic bijections, Ehrhart theory, and enumerative generating functions (Konstantinova et al., 2019, Fijavz et al., 2016, Orel, 2021, Iyama et al., 2022, Ferroni et al., 2022, Alain et al., 2010).

1. Generalized prisms in the Bubble-sort graph

In the Bubble-sort graph,

$$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$

a generalized prism is the Cartesian product $2\text{-}H=H\Box K_2$. The construction used in $BS_n$ takes $H=BS_{n-2}$ and defines

$GP_n:=2\text{-}BS_{n-2}\subset BS_n.$

For fixed distinct symbols $i,j\in\{1,\dots,n\}$, the subgraph $C_n\times P_2$0 is the induced subgraph whose vertices are precisely the permutations whose last two positions are $C_n\times P_2$1: $C_n\times P_2$2 It has two layers, one ending in $C_n\times P_2$3 and one ending in $C_n\times P_2$4, each a copy of $C_n\times P_2$5, while the vertical edges are exactly the edges induced by $C_n\times P_2$6, which swap the last two entries (Konstantinova et al., 2019).

As $C_n\times P_2$7 ranges over the $C_n\times P_2$8 unordered pairs, the subgraphs $C_n\times P_2$9 are vertex-disjoint and together cover every vertex of $\Gamma\overline{\Gamma}$0 exactly once; the paper describes this as a maximal cover. The factor graph whose vertices are these prisms and whose edges record adjacency between distinct prisms is precisely the Johnson graph $\Gamma\overline{\Gamma}$1. Since $\Gamma\overline{\Gamma}$2 is Hamiltonian, indeed Hamilton-connected, a Hamiltonian cycle in the factor graph can be lifted to a Hamiltonian cycle in $\Gamma\overline{\Gamma}$3: one traverses prisms in the order given by the factor-graph cycle and, inside each prism, joins the incoming and outgoing vertices by a Hamiltonian path. The key technical input is Proposition 3.2: in each $\Gamma\overline{\Gamma}$4, for any two vertices of opposite parity, there is a Hamiltonian path joining them. Because $\Gamma\overline{\Gamma}$5 is bipartite and the connecting $\Gamma\overline{\Gamma}$6-edges toggle parity, the required inside-prism paths always exist. In $\Gamma\overline{\Gamma}$7, this yields $\Gamma\overline{\Gamma}$8 prisms, each isomorphic to $\Gamma\overline{\Gamma}$9. This suggests that generalized prisms function as local Hamiltonian modules inside a larger Cayley graph (Konstantinova et al., 2019).

2. Prism graphs and threshold colorings

For Fijavž and Kriesell, the prism graph is $\Delta_n\times\Delta_1$0, with vertex set

$\Delta_n\times\Delta_1$1

two peripheral $\Delta_n\times\Delta_1$2-cycles, and spokes $\Delta_n\times\Delta_1$3. Given integers $\Delta_n\times\Delta_1$4 and a set $\Delta_n\times\Delta_1$5 of near edges, an $\Delta_n\times\Delta_1$6-threshold-coloring is a map $\Delta_n\times\Delta_1$7 such that

$\Delta_n\times\Delta_1$8

A graph is total threshold colorable if there exist integers $\Delta_n\times\Delta_1$9 such that for every $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$0, the graph admits such a coloring. The main theorem is that for every $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$1, the prism $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$2 is total threshold colorable, and in fact one can take $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$3 and $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$4 (Fijavz et al., 2016).

The proof proceeds by a case analysis on the arrangement of far edges around the $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$5 squares of the prism. If there is a useful cut of far edges splitting the prism into two ladder-like pieces, each piece can be $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$6-colored, producing an $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$7-coloring after reassembly. Otherwise, local structure in balanced or unbalanced 4-cycles is exploited, together with Lemma 2.2, to build a $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$8-coloring; an augmentation step then yields a uniform $$BS_n=\mathrm{Cay}\bigl(\mathrm{Sym}_n,\{b_1,b_2,\dots,b_{n-1}\}\bigr),\qquad b_i=(i\;i+1),$$9-coloring. Three auxiliary facts structure the argument: Lemma 2.1 is a 4-cycle sign lemma controlling relative inequalities in a mixed near/far square, Lemma 2.2 is the “A–B decomposition” that patches separate colorings through a small induced matching, and Lemma 3.1 shows that every ladder $2\text{-}H=H\Box K_2$0 is $2\text{-}H=H\Box K_2$1-total-threshold colorable (Fijavz et al., 2016).

The contrast with Möbius ladders is explicit: Möbius ladders are not total threshold colorable, and from this it follows that there is no characterization of being total threshold colorable in terms of a finite set of forbidden subgraphs. For particular near/far labelings, much smaller parameters can suffice. When all spokes of the triangular prism are near and all peripheral edges are far, a $2\text{-}H=H\Box K_2$2-coloring works; for the square prism with the same labeling pattern, the paper gives a concrete $2\text{-}H=H\Box K_2$3-coloring. The prism family therefore occupies a specific position in threshold-coloring theory: it is robust under arbitrary near/far edge partitions, but closely related cubic graphs need not share that robustness (Fijavz et al., 2016).

3. Complementary prisms, symmetry, and cores

Given a finite simple graph $2\text{-}H=H\Box K_2$4 on $2\text{-}H=H\Box K_2$5 vertices, its complementary prism $2\text{-}H=H\Box K_2$6 is obtained from $2\text{-}H=H\Box K_2$7 and its complement $2\text{-}H=H\Box K_2$8 by adding a perfect matching between corresponding vertices in the two layers. Writing

$2\text{-}H=H\Box K_2$9

one has $BS_n$0, and

$BS_n$1

Orel gives a general description of $BS_n$2. Outside the exceptional Petersen-type families $BS_n$3 and $BS_n$4, every automorphism either preserves the two layers and acts there by a single $BS_n$5, or swaps the layers and is induced by an antimorphism $BS_n$6. A corollary is that the ratio

$BS_n$7

can attain only the values $BS_n$8, $BS_n$9, $H=BS_{n-2}$0, and $H=BS_{n-2}$1 (Orel, 2021).

The same paper determines the Cheeger number: $H=BS_{n-2}$2 It also proves that $H=BS_{n-2}$3 is vertex-transitive if and only if $H=BS_{n-2}$4 is vertex-transitive and self-complementary. In this case, if $H=BS_{n-2}$5, the complementary prism is Hamiltonian-connected. At the same time, if $H=BS_{n-2}$6, then $H=BS_{n-2}$7 is never a Cayley graph (Orel, 2021).

The endomorphism theory is equally rigid. If $H=BS_{n-2}$8 is $H=BS_{n-2}$9-regular or vertex-transitive self-complementary with $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$0, then $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$1 is either the full graph, or isomorphic to $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$2 sitting in one layer, or isomorphic to $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$3 sitting in the other. From this, Orel proves that if $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$4 is strongly regular and self-complementary, then $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$5 is a core; likewise, if $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$6 is vertex-transitive self-complementary and either $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$7 is already a core or $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$8 is complete, then $GP_n:=2\text{-}BS_{n-2}\subset BS_n.$9 is a core. The model example is $i,j\in\{1,\dots,n\}$0: then $i,j\in\{1,\dots,n\}$1 is the Petersen graph, with $i,j\in\{1,\dots,n\}$2 vertices, $i,j\in\{1,\dots,n\}$3 edges, automorphism-group ratio $i,j\in\{1,\dots,n\}$4, Cheeger number $i,j\in\{1,\dots,n\}$5, vertex-transitivity, non-Cayley status, Hamiltonian-connectedness, and the core property (Orel, 2021).

4. Simplicial prisms and triangulations

The polytope prism $i,j\in\{1,\dots,n\}$6 is the product of the standard $i,j\in\{1,\dots,n\}$7-simplex

$i,j\in\{1,\dots,n\}$8

with the 1-simplex $i,j\in\{1,\dots,n\}$9. Its $C_n\times P_2$00 vertices are

$C_n\times P_2$01

Any choice of one vertex from each pair $C_n\times P_2$02 yields an $C_n\times P_2$03-simplex. If $C_n\times P_2$04, then

$C_n\times P_2$05

where $C_n\times P_2$06 if $C_n\times P_2$07 and $C_n\times P_2$08 if $C_n\times P_2$09. Such a simplex lies on the boundary exactly when $C_n\times P_2$10 or $C_n\times P_2$11; the internal $C_n\times P_2$12-simplices are therefore in bijection with the words in $C_n\times P_2$13 using both letters. In particular, there are $C_n\times P_2$14 internal $C_n\times P_2$15-simplices (Iyama et al., 2022).

The same paper identifies these internal simplices with indecomposable two-term presilting complexes over the preprojective algebra $C_n\times P_2$16 of type $C_n\times P_2$17. If $C_n\times P_2$18, then the $C_n\times P_2$19th coordinate of the $C_n\times P_2$20-vector is

$C_n\times P_2$21

and this determines the indecomposable presilting complex $C_n\times P_2$22. Thus

$C_n\times P_2$23

A triangulation $C_n\times P_2$24 of $C_n\times P_2$25 is a maximal collection of non-intersecting internal $C_n\times P_2$26-simplices, equivalently a basic two-term silting complex with $C_n\times P_2$27 indecomposable summands. The paper proves that triangulations are in bijection with permutations in $C_n\times P_2$28, hence

$C_n\times P_2$29

For $C_n\times P_2$30, the corresponding triangulation consists of the $C_n\times P_2$31 simplices

$C_n\times P_2$32

of which the internal ones are those with $C_n\times P_2$33 (Iyama et al., 2022).

Bistellar flips correspond exactly to silting mutations. If $C_n\times P_2$34 and $C_n\times P_2$35, then replacing the indecomposable summand $C_n\times P_2$36 by the unique different indecomposable presilting summand $C_n\times P_2$37 matches the replacement of one internal simplex by the unique other internal simplex in a minimal circuit. The symmetric group $C_n\times P_2$38 acts simply transitively on the set of triangulations by permuting the labels $C_n\times P_2$39, and the resulting graph of flips is the classical permutohedron of type $C_n\times P_2$40 of dimension $C_n\times P_2$41. The prism $C_n\times P_2$42 thus mediates a precise correspondence among triangulations, presilting objects, silting objects, mutations, and the symmetric group (Iyama et al., 2022).

5. Slices of rectangular prisms and Ehrhart theory

A different use of prisms arises from rectangular boxes and their hyperplane sections. Fix positive integers $C_n\times P_2$43 and an integer $C_n\times P_2$44, and define

$C_n\times P_2$45

together with the thin slice

$C_n\times P_2$46

These polytopes are described as generalizations of the hypersimplex; they are contained in the larger class of polypositroids introduced by Lam and Postnikov, and they coincide with polymatroids satisfying the strong exchange property up to an affinity. Fat slices $C_n\times P_2$47 are integrally equivalent to a thin slice in one higher dimension (Ferroni et al., 2022).

For the $C_n\times P_2$48-dimensional lattice polytope $C_n\times P_2$49, the Ehrhart polynomial

$C_n\times P_2$50

has nonnegative coefficients, and the paper gives a combinatorial formula for all of them in terms of $C_n\times P_2$51-compatible weighted permutations. If $C_n\times P_2$52 counts compatible weighted permutations with exactly $C_n\times P_2$53 cycles and total weight $C_n\times P_2$54, and $C_n\times P_2$55 is the Eulerian number, then

$C_n\times P_2$56

This formula proves Ehrhart positivity and gives a direct combinatorial interpretation of each coefficient (Ferroni et al., 2022).

The $C_n\times P_2$57-polynomial also has a combinatorial interpretation. Its coefficient $C_n\times P_2$58 counts $C_n\times P_2$59-compatible decorated ordered set partitions of type $C_n\times P_2$60 with winding number $C_n\times P_2$61. The leading Ehrhart coefficient, equivalently the normalized volume, admits the formula

$C_n\times P_2$62

where

$C_n\times P_2$63

The classical Laplace–Stanley identity $C_n\times P_2$64 appears as the special case $C_n\times P_2$65. The same theory identifies the graded algebra

$C_n\times P_2$66

with the Ehrhart ring of $C_n\times P_2$67, so its Hilbert function equals $C_n\times P_2$68 and its $C_n\times P_2$69-vector coincides with the prism-slice $C_n\times P_2$70-polynomial (Ferroni et al., 2022).

6. Inscribed polyominoes in a rectangular prism

Goupil and Cloutier study minimal 3D polyominoes inscribed in an $C_n\times P_2$71 rectangular prism. Such a polyomino is a finite set of unit cubes in $C_n\times P_2$72 that is face-connected, meets all six faces of the ambient prism, and has minimal volume under these conditions. Every such minimal polyomino has exactly

$C_n\times P_2$73

cubes. Equivalently, each of its three orthogonal projections onto the faces of the prism is a minimal 2D polyomino of area $C_n\times P_2$74, $C_n\times P_2$75, or $C_n\times P_2$76 (Alain et al., 2010).

Every minimal 3D inscribed polyomino falls into one of three disjoint families: diagonal polyominoes, formed from two hooks joined by a 3D stair along a prism diagonal; $C_n\times P_2$77 polyominoes, obtained by juxtaposing two perpendicular minimal 2D polyominoes with a short skew-hook; and skew-cross polyominoes, consisting of three mutually perpendicular minimal 2D corner-polyominoes meeting at a central cell of degree $C_n\times P_2$78. If

$C_n\times P_2$79

then

$C_n\times P_2$80

and each summand is a rational function. The 3D stair count is

$C_n\times P_2$81

while the skew-cross family has generating function

$C_n\times P_2$82

The theory also gives a recurrence for the number $C_n\times P_2$83 of corner-polyominoes and closed forms for subfamilies such as stairs, tripods, and corner-polyominoes (Alain et al., 2010).

Concrete enumerations illustrate the theory: $C_n\times P_2$84, $C_n\times P_2$85, and $C_n\times P_2$86. On the diagonal specialization $C_n\times P_2$87, the dominant singularity occurs at $C_n\times P_2$88, so

$C_n\times P_2$89

Taken together with prism slices, simplicial prisms, and graph prisms, these results suggest that prism constructions repeatedly convert geometric layering into explicit combinatorics: rational generating functions, cycle decompositions, Hamiltonian assemblies, and representation-theoretic parameterizations all emerge from the same two-level or sectioned architecture.