Topological and Algebraic Characterizations of Gallai-Simplicial Complexes

TL;DR

This paper explores the topological and algebraic properties of Gallai-simplicial complexes derived from planar graphs, computing Euler characteristics for specific graph classes and characterizing when certain graphs are Gallai graphs.

Contribution

It provides new formulas for Euler characteristics of Gallai-simplicial complexes and characterizes classes of graphs that are Gallai graphs based on their construction.

Findings

Euler characteristics computed for triangular ladder and prism graphs.

Identification of conditions under which specific graphs are Gallai graphs.

New algebraic and topological characterizations of Gallai-simplicial complexes.

Abstract

We recall first Gallai-simplicial complex $\Delta_{\Gamma}(G)$ associated to Gallai graph $\Gamma(G)$ of a planar graph $G$. The Euler characteristic is a very useful topological and homotopic invariant to classify surfaces. In Theorems 3.2 and 3.4, we compute Euler characteristics of Gallai-simplicial complexes associated to triangular ladder and prism graphs, respectively. Let $G$ be a finite simple graph on $n$ vertices of the form $n=3l+2$ or $3l+3$. In Theorem 4.4, we prove that $G$ will be $f$-Gallai graph for the following types of constructions of $G$. Type 1. When $n=3l+2$. $G=\mathbb{S}_{4l}$ is a graph consisting of two copies of star graphs $S_{2l}$ and $S'_{2l}$ with $l\geq 2$ having $l$ common vertices. Type 2. When $n=3l+3$. $G=\mathbb{S}_{4l+1}$ is a graph consisting of two star graphs $S_{2l}$ and $S_{2l+1}$ with $l\geq 2$ having $l$ common vertices.

Equations

$$\chi (X)=\sum _{k=0}^N(-1{)}^k{\beta }_k(X)$$

$$\dim \Delta =max\{\dim F|F\in \Delta \}.$$

$$\chi ({\Delta }_n)=1,\forall n\ge 0.$$

$$\chi (S^n)=2\chi ({\Delta }_n)-\chi (S^{n-1})=2-\chi (S^{n-1}).$$

$$2=\chi (S^n)+\chi (S^{n-1})=\dots =\chi (S^1)+\chi (S^0)$$

$$\chi ({\mathbb{R}}^n)=\chi ({\Delta }_n)-\chi (\partial {\Delta }_n)=1-\chi (S^{n-1})=(-1{)}^n.$$

$$\varnothing \subset {\Delta }^{(0)}\subset \dots \subset {\Delta }^{(N)}=\Delta$$

$$\chi (\Delta )=\chi ({\Delta }^{(0)})+\chi ({\Delta }^{(1)}\backslash {\Delta }^{(0)})+\dots +\chi ({\Delta }^{(N)}\backslash {\Delta }^{(N-1)}).$$

$$\chi (\Delta )=\sum _{k=0}^N(-1{)}^k{f}_k(\Delta ),$$

$${\Delta }_{\Gamma }(G)=<F|F\in \Omega (G)>,$$

$${\Delta }_{\Gamma }(G)=<\{2,4\},\{1,2,3\},\{1,3,4\},\{1,2,5\},\{1,4,5\}>.$$

$$E(L_n^{\ast })=\{e_{1,2},e_{2,3},\dots ,e_{2n-1,2n},e_{1,2n-1},e_{1,2n},\dots ,e_{n-1,n+1},e_{n-1,n+2}\}.$$

$$\chi ({\Delta }_{\Gamma }(L_n^{\ast }))=\sum _{k=0}^N(-1{)}^k{f}_k=0.$$

$$|\{l,j,k\}|=4+5(n-3)+2+(n-1)=6n-10$$

$$|\{j,k\}|=5+6(n-3)+4+2(n-1)+1=8n-10,$$

$$\chi ({\Delta }_{\Gamma }(L_n^{\ast }))=f_0-f_1+f_2=2n-(8n-10)+(6n-10)=0,$$

$$\chi ({\Delta }_{\Gamma }(Y_{3,n}))=\sum _{k=0}^N(-1{)}^k{f}_k=3(n-1).$$

$$f_2=7(n-2)+5(n-2)+3(n-2)+6+4+2=15n-18.$$

$$f_1=6(n-2)+5(n-2)+4(n-2)+5+4+3+2+1=15n-15.$$

$$\chi ({\Delta }_{\Gamma }(Y_{3,n}))=f_0-f_1+f_2=3n-(15n-15)+(15n-18)=3(n-1),$$

Full text

Let $G$ be a finite simple graph on $n$ vertices of the form $n=3l+2$ or $3l+3$. In Theorem 4.4, we prove that $G$ will be $f$-Gallai graph for the following types of constructions of $G$.

Type 1. When $n=3l+2$. $G=\mathbb{S}_{4l}$ is a graph consisting of two copies of star graphs $S_{2l}$ and $S^{\prime}_{2l}$ with $l\geq 2$ having $l$ common vertices.

Type 2. When $n=3l+3$. $G=\mathbb{S}_{4l+1}$ is a graph consisting of two star graphs $S_{2l}$ and $S_{2l+1}$ with $l\geq 2$ having $l$ common vertices.

Key words: Euler characteristic, simplicial complex and $f$-ideals.

*2010 Mathematics Subject Classification: Primary 05E25, 55U10, 13P10 Secondary 06A11, 13H10.

1. Introduction

Let $X$ be a finite CW complex of dimension $N$. The Euler characteristic is a function $\chi$ which associates to each $X$ an integer $\chi(X)$. More explicitly, the Euler characteristic of $X$ is defined as the alternating sum

[TABLE]

with $\beta_{k}(X)=rank(H_{k}(X))$ the $k$-th Betti number of $X$.

The Euler characteristic is a very useful topological and homotopic invariant to classify surfaces. The Euler characteristic is uniquely determined by excision $\chi(X)=\chi(C)+\chi(X\backslash C)$, for every closed subset $C\subset X$. The excision property has a dual form $\chi(X)=\chi(U)+\chi(X\backslash U)$, for every open subset $U\subset X$, see [7] and [10] for more details.

We consider a planar graph $G$, the Gallai graph $\Gamma(G)$ of $G$ is a graph having edges of $G$ as its vertices, that is, $V(\Gamma(G))=E(G)$ and two distinct edges of $G$ are adjacent in $\Gamma(G)$ if they are adjacent in $G$ but dot span a triangle. The buildup of the 2-dimensional Gallai-simplicial complex $\Delta_{\Gamma}(G)$ from a planar graph $G$ is an abstract idea similar to building an origami shape from a plane sheet of paper by defining a crease pattern, see [2].

Let $S=k[x_{1},\ldots,x_{n}]$ be a polynomial ring over an infinite field $k$. There is a natural bijection between a square free monomial ideal and a simplicial complex written as $\Delta\leftrightarrow I_{\mathcal{N}}(\Delta)$, where $I_{\mathcal{N}}(\Delta)$ is the Stanley-Reisner ideal or non-face ideal of $\Delta$, see for instance [4]. In [5], Faridi introduced another correspondence $\Delta\leftrightarrow I_{\mathcal{F}}(\Delta)$, where $I_{\mathcal{F}}(\Delta)$ is the facet ideal of $\Delta$. She discussed its connections with the theory of Stanley-Reisner rings.

In [1] and [3], the authors investigated the correspondence $\delta_{\mathcal{F}}(I)\leftrightarrow I\leftrightarrow\delta_{\mathcal{N}}(I)$, where $\delta_{\mathcal{F}}(I)$ and $\delta_{\mathcal{N}}(I)$ are facet and non-face simplicial complexes associated to the square free monomial ideal $I$ (respectively). A square free monomial ideal $I$ in $S$ is said to be an $f$-ideal if and only if both $\delta_{\mathcal{F}}(I)$ and $\delta_{\mathcal{N}}(I)$ have the same $f$-vector. The concepts of $f$- ideals is important in the sense that it discovers new connections between both the theories. The complete characterization of $f$-ideals in the polynomial ring $S$ over a field $k$ can be found in [3]. A simple finite graph $G$ is said to be the $f$-graph if its edge ideal $I(G)$ is an $f$-ideal of degree 2, see [9].

In Theorems 3.2 and 3.4, we compute Euler characteristics of Gallai-simplicial complexes associated to triangular ladder and prism graphs, respectively.

Let $G$ be a finite simple graph on $n$ vertices of the form $n=3l+2$ or $3l+3$. In Theorem 4.4, we prove that $G$ will be $f$-Gallai graph for the following types of constructions of $G$.

Type 1. When $n=3l+2$. $G=\mathbb{S}_{4l}$ is a graph consisting of two copies of star graphs $S_{2l}$ and $S^{\prime}_{2l}$ with $l\geq 2$ having $l$ common vertices.

Type 2. When $n=3l+3$. $G=\mathbb{S}_{4l+1}$ is a graph consisting of two star graphs $S_{2l}$ and $S_{2l+1}$ with $l\geq 2$ having $l$ common vertices.

2. Preliminaries

A simplicial complex $\Delta$ on $[{n}]=\{1,\ldots,n\}$ is a collection of subsets of $[n]$ with the property that $\{i\}\in\Delta$ for all $i$, and if $F\in\Delta$ then every subset of $F$ will belong to $\Delta$ (including empty set). The elements of $\Delta$ are called faces of $\Delta$ and the dimension of a face $F\in\Delta$ is defined as $|F|-1$, where $|F|$ is the number of vertices of $F$. The faces of dimensions [math] and $1$ are called vertices and edges, respectively, and $dim\ \emptyset=-1$.

The maximal faces of $\Delta$ under inclusion are called facets. The dimension of $\Delta$ is denoted by $\dim\Delta$ and is defined as:

[TABLE]

A simplicial complex is said to be pure if it has all the facets of the same dimension. If $\{F_{1},\ldots,F_{q}\}$ is the set of all the facets of $\Delta$, then $\Delta=<F_{1},\ldots,F_{q}>$.

We denote by $\Delta_{n}$ the closed $n$-dimensional simplex. Every simplex $\Delta_{n}$ is homotopic to a point and thus

[TABLE]

Note that $\partial\Delta_{n}$ is homeomorphic to the $(n-1)$-sphere $S^{n-1}$. Since $S^{0}$ is a union of two points, we have $\chi(S^{0})=2$. In general, the $n$-dimensional sphere is a union of two closed hemispheres intersecting along the Equator which is a $(n-1)$ sphere. Therefore,

[TABLE]

We deduce inductively

[TABLE]

so that $\chi(S^{n})=1+(-1)^{n}$. Now, note that the interior of $\Delta_{n}$ is homeomorphic to $\mathbb{R}^{n}$ so that

[TABLE]

The excision property implies the following useful formula. Suppose

[TABLE]

is an increasing filtration of $\Delta$ by closed subsets. Then,

[TABLE]

We denote by $\Delta^{(k)}$ the union of the simplices of dimension $\leq k$. Then, $\Delta^{(k)}\backslash\Delta^{(k-1)}$ is the union of interiors of the $k$-dimensional simplices. We denote by $f_{k}(\Delta)$ the number of such simplices. Each of them is homeomorphic to $\mathbb{R}^{k}$ and thus its Euler characteristic is equal to $(-1)^{k}$. Consequently, the Euler characteristic of $\Delta$ is given by

[TABLE]

see [7] and [10].

Let $\Delta$ be a simplicial complex of dimension $N$, we define its $f$-vector by a $(N+1)$-tuple $f=(f_{0},\ldots,f_{N})$, where $f_{i}$ is the number of $i$-dimensional faces of $\Delta$.

The following definitions serve as a bridge between the combinatorial and algebraic properties of the simplicial complexes over the finite set of vertices $[n]$.

Let $\Delta$ be a simplicial complex over the vertex set $\{v_{1},\ldots,v_{n}\}$ and $S=k[x_{1},\ldots,x_{n}]$ be the polynomial ring on $n$ variables. We define the facet ideal of $\Delta$ by $I_{\mathcal{F}}(\Delta)$, which is an ideal of $S$ generated by square free monomials $x_{i_{1}}\ldots x_{i_{s}}$ where $\{v_{i_{1}},\ldots,v_{i_{s}}\}$ is a facet of $\Delta$. We define the non-face ideal or the Stanley-Reisner ideal of $\Delta$ by $I_{\mathcal{N}}(\Delta)$, which is an ideal of $S$ generated by square free monomials $x_{i_{1}}\ldots x_{i_{s}}$ where $\{v_{i_{1}},\ldots,v_{i_{s}}\}$ is a non-face of $\Delta$.

Let $I=(M_{1},\dots,M_{q})$ be a square free monomial ideal in the polynomial ring $S=k[x_{1},\ldots,x_{n}]$, where $\{M_{1},\dots,M_{q}\}$ is a minimal generating set of $I$. We define a simplicial complex $\delta_{\mathcal{F}}(I)$ over a set of vertices $v_{1},\ldots,v_{n}$ with facets $F_{1},\dots,F_{q}$, where for each $i$, $F_{i}=\{v_{j}\ |\ x_{j}|M_{i},1\leq j\leq n\}$. $\delta_{\mathcal{F}}(I)$ is said to be the facet complex of $I$. We define a simplicial complex $\delta_{\mathcal{N}}(I)$ over a set of vertices $v_{1},\ldots,v_{n}$, where $\{v_{i_{1}},\ldots,v_{i_{s}}\}$ a face of $\delta_{\mathcal{N}}(I)$ if and only if the product $x_{i_{1}}\ldots x_{i_{s}}$ does not belong to $I$. We call $\delta_{\mathcal{N}}(I)$ the non-face complex or the Stanley-Reisner complex of $I$.

To proceed further, we define the Gallai-graph $\Gamma(G)$, which is a nice combinatorial buildup, see [6] and [8].

Definition 2.1**.**

Let $G$ be a graph and $\Gamma(G)$ is said to be the Gallai graph of $G$ if the following conditions hold;

1. Each edge of $G$ represents a vertex of $\Gamma(G)$.

2. If two edges are adjacent in $G$ that do not span a triangle in $G$ then their corresponding vertices will be adjacent in $\Gamma(G)$.**

Example 2.2**.**

The graph $G$ and its Gallai graph $\Gamma(G)$ are given in figures (i) and (ii), repectively*.*

To define Gallai-simplicial complex $\Delta_{\Gamma}(G)$ of a planar graph $G$, we introduce first a few notions, see [2].

Definition 2.3**.**

[2]** Let $G$ be a finite simple graph with vertex set $V(G)=[n]$ and edge set $E(G)=\{e_{i,j}=\{i,j\}\,|\,i,j\in V(G)\}$. We define the set of Gallai-indices $\Omega(G)$ of the graph $G$ as the collection of subsets of $V(G)$ such that if $e_{i,j}$ and $e_{j,k}$ are adjacent in $\Gamma(G)$, then $F_{i,j,k}=\{i,j,k\}\in\Omega(G)$ or if $e_{i,j}$ is an isolated vertex in $\Gamma(G)$ then $F_{i,j}=\{i,j\}\in\Omega(G)$.

Definition 2.4**.**

[2*]** *A Gallai-simplicial complex $\Delta_{\Gamma}(G)$ of $G$ is a simplicial complex defined over $V(G)$ such that

[TABLE]

where $\Omega(G)$ is the set of Gallai-indices of $G$.**

Example 2.5**.**

For the graph $G$ shown in figure below, its Gallai-simplicial complex $\Delta_{\Gamma}(G)$ is given by**

[TABLE]

3. Characterizations of Gallai-Simplicial Complexes

The ladder graph $L_{n}$ is a planar undirected graph with $2n$ vertices and $3n-2$ edges. The ladder graph $L_{n}$ is the cartesian product of two path graphs $P_{n}$ and $P_{2}$, that is $L_{n}=P_{n}\times P_{2}$ and looks like a ladder with $n$ rungs. The path graph $P_{n}$ is a graph whose vertices can be listed in an order $v_{1},\ldots,v_{n}$ such that $\{v_{i},v_{i+1}\}$ is an edge for $1\leq i\leq n-1$. If we add a cross edge between every two consecutive rungs of the ladder then the resulting graph is said to be a triangular ladder graph $L^{\ast}_{n}$ with $2n$ vertices and $4n-3$ edges.

Lemma 3.1**.**

Let $L^{\ast}_{n}$ be the triangular ladder graph on $2n$ vertices with fixing the label of the edge-set $E(L^{\ast}_{n})$ as follows;

[TABLE]

*Then, we have

$\Omega(L^{\ast}_{n})=\{F_{1,2,3},\ldots,F_{n-2,n-1,n},F_{n,n+1,n+2},\ldots,F_{2n-2,2n-1,2n},\\ F_{1,2,2n},F_{2,3,2n-1},\ldots,F_{n-1,n,n+2},F_{1,2,2n-2},\ldots,F_{n-2,n-1,n+1},\\ F_{1,2n-2,2n-1},\ldots,F_{n-2,n+1,n+2},F_{2,2n-1,2n},\ldots,F_{n-1,n+2,n+3}\}.$*

Proof.

By definition, it is clear that $F_{i,i+1,i+2}\in\Omega(L^{\ast}_{n})$ because $i$, $i+1$, $i+2$ are consecutive vertices of $2n$-cycle and edges $e_{i,i+1}$ and $e_{i+1,i+2}$ do not span a triangle except $F_{n-1,n,n+1}$ and $F_{2n-1,2n,1}$ as the edge sets $\{e_{n-1,n},e_{n,n+1}\}$ and $\{e_{2n-1,2n},e_{2n,1}\}$ span triangles in the triangular ladder graph $L^{\ast}_{n}$. Moreover, $F_{i,i+1,j}\in\Omega(L^{\ast}_{n})$ for indices of types $1\leq i\leq n-1$; $j=2n+1-i$ and $1\leq i\leq n-2$; $j=2n-1-i$. Also, $F_{i,j,j+1}\in\Omega(L^{\ast}_{n})$ for indices of types $1\leq i\leq n-2$; $j=2n-1-i$ and $2\leq i\leq n-1$; $j=2n+1-i$. Hence the result.

Theorem 3.2**.**

Let $\Delta_{\Gamma}(L^{\ast}_{n})$ be the Gallai simplicial complex of triangular ladder graph $L^{\ast}_{n}$ with $2n$ vertices for $n\geq 3$. Then, the Euler characteristic of $\Delta_{\Gamma}(L^{\ast}_{n})$ is

[TABLE]

Proof.

Since, the triangular ladder graph has $2n$ vertices therefore, we have $f_{0}=2n$.

Moreover, for $\{l,j,k\}\in\Delta_{\Gamma}(L^{\ast}_{n})$ with $1\leq l\leq 2n-2$ and $j,k\in[2n]$, we have

1. (1)

$|\{1,j,k\}|=4$ with $\{j,k\}\in\{\{2,3\},\{2,2n-2\},\{2,2n\},\{2n-2,2n-1\}\}$; 2. (2)

$|\{l,j,k\}|=5(n-3)$ for $2\leq l\leq n-2$ and $\{j,k\}\in\{\{l+1,l+2\},\{l+1,2n-1-l\},\{l+1,2n+1-l\},\{2n-1-l,2n-l\},\{2n+1-l,2n+2-l\}\}$; 3. (3)

$|\{n-1,j,k\}|=2$ with $\{j,k\}\in\{\{n,n+2\},\{n+2,n+3\}\}$; 4. (4)

$|\{l,l+1,l+2\}|=n-1$ for $n\leq l\leq 2n-2$.

Adding the results from $(1)$ to $(4)$, we get

[TABLE]

with $1\leq l\leq 2n-2$ and $j,k\in[2n]$. Therefore, $f_{2}=6n-10$.

Now, for $\{j,k\}\in\Delta_{\Gamma}(L^{\ast}_{n})$ with $1\leq j\leq 2n-1$ and $k\in[2n]$, we have 5. (5)

$|\{1,k\}|=5$, where $k\in\{2,3,2n-2,2n-1,2n\}$; 6. (6)

$|\{j,k\}|=6(n-3)$ with $2\leq j\leq n-2$ and $k\in\{j+1,j+2,2n-1-j,2n-j,2n+1-j,2n+2-j\}$; 7. (7)

$|\{n-1,k\}|=4$, where $k\in\{n,n+1,n+2,n+3\}$; 8. (8)

$|\{j,k\}|=2(n-1)$ with $n\leq j\leq 2n-2$ and $k\in\{j+1,j+2\}$; 9. (9)

$|\{2n-1,2n\}|=1$.

Adding the results from $(5)$ to $(9)$, we obtain

[TABLE]

where $1\leq j\leq 2n-1$ and $k\in[2n]$. Therefore, $f_{1}=8n-10$.

Thus, we compute

[TABLE]

which is the desired result.

The Prism graph $Y_{3,n}$ is a simple graph defined by the cartesian product $Y_{3,n}=C_{3}\times P_{n}$ with $3n$ vertices and $3(2n-1)$ edges. We label the edge-set of $Y_{3,n}$ in the following way;

$E(Y_{3,n})=\{e_{1,2},e_{2,3},e_{3,1},e_{4,5},e_{5,6},e_{6,4},\ldots,e_{3i+1,3i+2},e_{3i+2,3i+3},e_{3i+3,3i+1},\ldots,\\ e_{3n-2,3n-1},e_{3n-1,3n},e_{3n,3n-2},e_{1,4},e_{4,7},\ldots,e_{3n-5,3n-2},e_{2,5},e_{5,8},\ldots,e_{3n-4,3n-1},e_{3,6},\\ e_{6,9},\ldots,e_{3n-3,3n}\}$, where $e_{3i+1,3i+2},e_{3i+2,3i+3},e_{3i+3,3i+1}$ for $0\leq i\leq n-1$ are the edges of $(i+1)$-$th$ $C_{3}$-cycle.

Lemma 3.3**.**

*Let $Y_{3,n}$ be a prism graph on the vertex set $[3n]$ and edge set $E(Y_{3,n})$, with labeling of edges given above. Then, we have

$\Omega(Y_{3,n})=\{F_{1,2,4},F_{1,2,5},F_{2,3,5},F_{2,3,6},F_{4,5,1},F_{4,5,2},F_{4,5,7},F_{4,5,8},F_{5,6,2},F_{5,6,3},F_{5,6,8},\\ F_{5,6,9},\ldots,F_{3n-5,3n-4,3n-8},F_{3n-5,3n-4,3n-7},F_{3n-5,3n-4,3n-2},F_{3n-5,3n-4,3n-1},\\ F_{3n-4,3n-3,3n-7},F_{3n-4,3n-3,3n-6},F_{3n-4,3n-3,3n-1},F_{3n-4,3n-3,3n},F_{3n-2,3n-1,3n-5},\\ F_{3n-2,3n-1,3n-4},F_{3n-1,3n,3n-4},F_{3n-1,3n,3n-3},F_{3,1,6},F_{3,1,4},F_{6,4,3},F_{6,4,1},F_{6,4,9},F_{6,4,7},\\ \ldots,F_{3n-3,3n-5,3n-6},F_{3n-3,3n-5,3n-8},F_{3n-3,3n-5,3n},F_{3n-3,3n-5,3n-2},F_{3n,3n-2,3n-3},\\ F_{3n,3n-2,3n-5},F_{1,4,7},\ldots,F_{3n-8,3n-5,3n-2},F_{2,5,8},\ldots,F_{3n-7,3n-4,3n-1},F_{3,6,9},\\ \ldots,F_{3n-6,3n-3,3n}\}$.*

Proof.

By definition, one can easily see that $F_{3i+1,3i+2,3i+3}$ does not belong to $\Omega(Y_{3,n})$ because $3i+1,3i+2,3i+3$ with $0\leq i\leq n-1$ are vertices of $(i+1)$-$th$ $C_{3}$-cycle. Therefore, from construction of all possible triangles in prism graph $Y_{3,n}$, we have

$(i)$ $F_{j,j+1,j-3},F_{j,j+1,j-2}\in\Omega(Y_{3,n})$ for $4\leq j\leq 3n-1$ but $j$ is not multiple of 3;

$(ii)$ $F_{j,j+1,j+3},F_{j,j+1,j+4}\in\Omega(Y_{3,n})$ for $1\leq j\leq 3n-4$ but $j$ is not multiple of 3;

$(iii)$ $F_{3j,3j-2,3j-3},F_{3j,3j-2,3j-5}\in\Omega(Y_{3,n})$ for $2\leq j\leq n$;

$(iv)$ $F_{3j,3j-2,3j+3},F_{3j,3j-2,3j+1}\in\Omega(Y_{3,n})$ for $1\leq j\leq n-1$;

$(v)$ $F_{j,j+3,j+6}\in\Omega(Y_{3,n})$ for $1\leq j\leq 3n-6$.

Hence the proof.

Theorem 3.4**.**

Let $\Delta_{\Gamma}(Y_{3,n})$ be the Gallai-simplicial complex of prism graph $Y_{3,n}$ with $3n$ vertices for $n\geq 3$. Then, the Euler characteristic of $\Delta_{\Gamma}(Y_{3,n})$ is

[TABLE]

Proof.

Since, the prism graph has $3n$ vertices therefore, we have $f_{0}=3n$.

Now, for $\{3l+i,j,k\}\in\Delta_{\Gamma}(Y_{3,n})$ with $0\leq l\leq n-2$ and $j,k\in[3n]$ such that $i=1,2,3$, we have

1. (1)

$|\{3l+1,j,k\}|=7(n-2)$ with $0\leq l\leq n-3$ and $\{j,k\}\in\{\{3l+2,3l+4\},\{3l+2,3l+5\},\{3l+3,3l+4\},\{3l+3,3l+6\},\{3l+4,3l+5\},\{3l+4,3l+6\},\{3l+4,3l+7\}\}$; 2. (2)

$|\{3l+2,j,k\}|=5(n-2)$ for $0\leq l\leq n-3$ and $\{j,k\}\in\{\{3l+3,3l+5\},\{3l+3,3l+6\},\{3l+4,3l+5\},\{3l+5,3l+6\},\{3l+5,3l+8\}\}$; 3. (3)

$|\{3l+3,j,k\}|=3(n-2)$ for $0\leq l\leq n-3$ and $\{j,k\}\in\{\{3l+4,3l+6\},\{3l+5,3l+6\},\{3l+6,3l+9\}\}$; 4. (4)

$|\{3n-5,j,k\}|=6$, where $\{j,k\}\in\{\{3n-4,3n-2\},\{3n-4,3n-1\},\{3n-3,3n-2\},\{3n-3,3n\},\{3n-2,3n-1\},\{3n-2,3n\}\}$; 5. (5)

$|\{3n-4,j,k\}|=4$, where $\{j,k\}\in\{\{3n-3,3n-1\},\{3n-3,3n\},\{3n-2,3n-1\},\{3n-1,3n\}\}$; 6. (6)

$|\{3n-3,j,k\}|=2$, where $\{j,k\}\in\{\{3n-2,3n\},\{3n-1,3n\}\}$.

Adding the results from $(1)$ to $(6)$, we get

[TABLE]

Next, for $\{3j+i,k\}\in\Delta_{\Gamma}(Y_{3,n})$ with $0\leq j\leq n-2$ and $k\in[3n]$ such that $i=1,2,3$, we obtain 7. (7)

$|\{3j+1,k\}|=6(n-2)$ with $0\leq j\leq n-3$ and $k\in\{3j+2,3j+3,3j+4,3j+5,3j+6,3j+7\}$; 8. (8)

$|\{3j+2,k\}|=5(n-2)$ with $0\leq j\leq n-3$ and $k\in\{3j+3,3j+4,3j+5,3j+6,3j+8\}$; 9. (9)

$|\{3j+3,k\}|=4(n-2)$ with $0\leq j\leq n-3$ and $k\in\{3j+4,3j+5,3j+6,3j+9\}$; 10. (10)

$|\{3n-5,k\}|=5$, where $k\in\{3n-4,3n-3,3n-2,3n-1,3n\}$; 11. (11)

$|\{3n-4,k\}|=4$, where $k\in\{3n-3,3n-2,3n-1,3n\}$; 12. (12)

$|\{3n-3,k\}|=3$, where $k\in\{3n-2,3n-1,3n\}$.

Moreover, we have 13. (13)

$|\{3n-2,k\}|=2$, where $k\in\{3n-1,3n\}$; 14. (14)

$|\{3n-1,3n\}|=1.$

Adding the results from $(7)$ to $(14)$, we get

[TABLE]

Hence, we compute

[TABLE]

which is the desired result.

4. Construction of $f$-Gallai graphs

We introduce first the $f$-Gallai graph.

Definition 4.1**.**

A finite simple graph $G$ is said to be $f$-Gallai graph, if the edge ideal $I(\Gamma(G))$ of the Gallai graph $\Gamma(G)$ is an $f$-ideal*.*

The following theorem provided us a construction of $f$-graphs.

Theorem 4.2**.**

[9*]**. *Let $G$ be a simple graph on $n$ vertices. Then for the following constructions, $G$ will be $f$-graph:

Case(i) When $n=4l$. $G$ consists of two components $G_{1}$ and $G_{2}$ joined with $l$-edges, where both $G_{1}$ and $G_{2}$ are the complete graphs on $2l$ vertices.

Case(ii) When $n=4l+1$. $G$ consists of two components $G_{1}$ and $G_{2}$ joined with $l$-edges, where $G_{1}$ is the complete graph on $2l$ vertices and $G_{2}$ is the complete graph on $2l+1$ vertices.**

Definition 4.3**.**

The star graph $S_{n}$ is a complete bipartite graph $K_{1,n}$ on $n+1$ vertices and $n$ edges formed by connecting a single vertex (central vertex) to all other vertices.**

We establish now the following result.

Theorem 4.4**.**

Let $G$ be a finite simple graph on $n$ vertices of the form $n=3l+2$ or $3l+3$. Then for the following constructions, $G$ will be $f$-Gallai graph.

Type 1. When $n=3l+2$. $G=\mathbb{S}_{4l}$ is a graph consisting of two copies of star graphs $S_{2l}$ and $S^{\prime}_{2l}$ with $l\geq 2$ having $l$ common vertices.

Type 2. When $n=3l+3$. $G=\mathbb{S}_{4l+1}$ is a graph consisting of two star graphs $S_{2l}$ and $S_{2l+1}$ with $l\geq 2$ having $l$ common vertices.**

Proof.

Type 1. When $n=3l+2$, the number of edges in $\mathbb{S}_{4l}$ will be $4l$, as shown in figure $\mathbb{S}_{12}$ with l=3. Let $\{e_{1},\ldots,e_{2l}\}$ and $\{e^{\prime}_{1},\ldots,e^{\prime}_{2l}\}$ be the edge sets of the star graphs $S_{2l}$ and $S^{\prime}_{2l}$, respectively such that $e_{i}$ and $e^{\prime}_{i}$ have a common vertex for each $i=1,\ldots,l$. While finding Gallai graph $\Gamma(\mathbb{S}_{4l})$ of the graph $\mathbb{S}_{4l}$, we observe that the edges $e_{1},\ldots,e_{2l}$ of the star graph $S_{2l}$ in $\mathbb{S}_{4l}$ will induce a complete graph $\Gamma(S_{2l})$ on $2l$ vertices in the Gallai graph $\Gamma(\mathbb{S}_{4l})$, as shown in figure $\Gamma(\mathbb{S}_{12})$ with $l=3$. Similarly, the edges $e^{\prime}_{1},\ldots,e^{\prime}_{2l}$ of the star graph $S^{\prime}_{2l}$ will induce another complete graph $\Gamma(S^{\prime}_{2l})$ on $2l$ vertices in $\Gamma(\mathbb{S}_{4l})$. As, $e_{i}$ and $e^{\prime}_{i}$ are the adjacent edges in $\mathbb{S}_{4l}$ for each $i=1,\ldots,l$. Therefore, $e_{i}$ and $e^{\prime}_{i}$ will be incident vertices in $\Gamma(\mathbb{S}_{4l})$ for every $i=1,\ldots,l$. Thus, Gallai graph $\Gamma(\mathbb{S}_{4l})$ having $4l$ vertices consists of two components $\Gamma(S_{2l})$ and $\Gamma(S^{\prime}_{2l})$ joined with $l$- edges, where both $\Gamma(S_{2l})$ and $\Gamma(S^{\prime}_{2l})$ are complete graphs on $2l$ vertices. Therefore, by Theorem 4.2, $\Gamma(\mathbb{S}_{4l})$ is $f$-Gallai graph.

Type 2. When $n=3l+3$, the number of edges in $\mathbb{S}_{4l+1}$ will be $4l+1$, see figure $\mathbb{S}_{13}$ (where $l=3$). Let $\{e_{1},\ldots,e_{2l}\}$ and $\{e^{\prime}_{1},\ldots,e^{\prime}_{2l+1}\}$ be the edge sets of the star graphs $S_{2l}$ and $S_{2l+1}$ (respectively) such that $e_{i}$ and $e^{\prime}_{i}$ share a common vertex for each $i=1,\ldots,l$. One can easily see that the edges $e_{1},\ldots,e_{2l}$ of $S_{2l}$ in $\mathbb{S}_{4l+1}$ will induce a complete graph $\Gamma(S_{2l})$ on $2l$ vertices in the Gallai graph $\Gamma(\mathbb{S}_{4l+1})$, see figure $\Gamma(\mathbb{S}_{13})$ (where $l=3$). Similarly, the edges $e^{\prime}_{1},\ldots,e^{\prime}_{2l+1}$ of $S_{2l+1}$ will induce another complete graph $\Gamma(S_{2l+1})$ on $2l+1$ vertices in $\Gamma(\mathbb{S}_{4l+1})$. Since $e_{i}$ and $e^{\prime}_{i}$ are the adjacent edges in $\mathbb{S}_{4l+1}$ for every $i=1,\ldots,l$. Therefore, $e_{i}$ and $e^{\prime}_{i}$ will be incident vertices in the Gallai graph $\Gamma(\mathbb{S}_{4l+1})$ for each $i=1,\ldots,l$. Thus, the Gallai graph $\Gamma(\mathbb{S}_{4l+1})$ having $4l+1$ vertices consists of two components $\Gamma(S_{2l})$ and $\Gamma(S_{2l+1})$ joined with $l$-edges, where $\Gamma(S_{2l})$ and $\Gamma(S_{2l+1})$ are complete graphs on $2l$ and $2l+1$ vertices, respectively. Hence, by Theorem 4.2, $\Gamma(\mathbb{S}_{4l+1})$ is $f$-Gallai graph.

Remark 4.5**.**

One can easily see that the Gallai graph of the line graph $L_{n}$ is isomorphic to $L_{n-1}$ and that of cyclic graph $C_{n}$ is isomorphic to $C_{n}$. Therefore, both $\Gamma(L_{n})$ and $\Gamma(C_{n})$ are $f$-Gallai graphs if and only if $n=5$, see [9].