Formulas for the Number of Weak Homomorphisms from Paths to Rectangular Grid Graphs

Abstract

A weak homomorphism from graph G to graph H is a mapping $f:V\left(G\right)\to V\left(H\right)$, where either $f\left(x\right)=f\left(y\right)$ or $\left\{f\right(x),f(y\left)\right\}\in E\left(H\right)$ hold for all $\{x,y\}\in E\left(G\right)$. A rectangular grid graph is formed by taking the Cartesian product of two paths. Counting weak homomorphisms is a fundamental problem in graph theory. In this paper, we present formulas for calculating the number of weak homomorphisms from paths to rectangular grid graphs. This count directly corresponds to the number of partial walks with length m within the rectangular grid graph, offering a combinatorial solution to this enumeration problem.

1. Introduction

In mathematics, the image refers to the set of values obtained by applying a mapping to all elements within the domain. Within this image, certain structural properties of the domain are retained. A mapping that maintains such a structure, which is of particular interest for our study, is commonly referred to as a homomorphism. In the context of graphs, a homomorphism is defined as follows.

Consider graphs G and H. A mapping, denoted as $f:V\left(G\right)\to V\left(H\right),$ is a homomorphism from G to H if $\left\{f\right(x),f(y\left)\right\}\in E\left(H\right)$ for all $\{x,y\}\in E\left(G\right)$, meaning that f preserves the edges. The set of homomorphisms from G to H is denoted as $\mathrm{Hom}(G,H)$. An endomorphism on G is a homomorphism from G to itself, and the set of endomorphisms on G is denoted by $\mathrm{End}\left(G\right)$. Let $P_n$ represent a path of order n with vertex set $V\left(P_n\right)=\{0,1,\dots ,n-1\}$ and edge set $E\left(P_n\right)=\left\{\{i,i+1\}\right|i=0,1,\dots ,n-2\}$. Similarly, $C_n$ stands for a cycle of order n ($n\ge 3$) with vertices $V\left(C_n\right)=\{0,1,\dots ,n-1\}$ and edges $E\left(C_n\right)=\left\{\{i,i+1\}\right|i=0,1,\dots ,n-1\}$, where the addition is performed modulo n. For a deeper understanding of graphs and algebraic graphs, we direct readers to references [1,2].

Graph homomorphisms, and specifically endomorphisms, have been the subject of much research. Böttcher and Knauer (1992) [3] detailed various ways to define graph homomorphisms, which resulted in six categories of endomorphisms for any given graph: endomorphisms, half-strong endomorphisms, locally strong endomorphisms, quasi-strong endomorphisms, strong endomorphisms, and automorphisms. It is evident that the set of endomorphisms on G constitutes a monoid, wherein the composition of mappings serves as the defining operation. The formulas used to count graph homomorphisms and endomorphisms are essential for analyzing the structure of $\mathrm{Hom}(G,H)$ and $\mathrm{End}\left(G\right)$, respectively. The expression for determining the number of endomorphisms on $P_n$ was introduced by Arworn in 2009 [4]. Arworn transformed the problem by equating it to enumerating the shortest paths originating from point $(0,0)$ and reaching any point $(i,j)$ within an r-ladder square lattice, ultimately deriving a succinct formula. Arworn and Wojtylak [5] also presented a formula in the same year to calculate the number of homomorphisms from path $P_m$ to path $P_n$. This formula uses the order of the set ${\mathrm{Hom}}_j^i(P_m,P_n)$, where ${\mathrm{Hom}}_j^i(P_m,P_n)=\{f\in \mathrm{Hom}(P_m,P_n)\mid f\left(0\right)=i,f(m-1)=j\}$ for all $i,j\in \{0,1,\dots ,n-1\}$. In 2014, Eggleton and Morayne [6] provided identities and formulas for counting homomorphisms of a finite path into a finite path. The paper also presents closed-form formulas for the number of endomorphisms of a finite path. In the same year, Csikvari and Lin [7] studied the number of homomorphisms from a tree with m vertices ($T_m$) to any graph G. They established a lower bound for the number of these homomorphisms in terms of the largest eigenvalue of the adjacency matrix of G. In 2015, Csikvari and Lin [8] further investigated the number of homomorphisms of trees into a path, proving the inequality:

$$|\mathrm{Hom}(P_m,P_n)|\hspace{0.17em}\le \hspace{0.17em}|\mathrm{Hom}(T_m,P_n)|\hspace{0.17em}\le \hspace{0.17em}|\mathrm{Hom}(S_m,P_n)|$$

where $T_m$ is any tree on m vertices and $S_m$ denotes the star on m vertices.

The concept of counting standard homomorphisms between graphs has been well studied. However, many problems require a more general mapping, one that allows for the collapsing of edges. This need motivates the introduction of weak homomorphisms, which extend the traditional definition by allowing the edges to be contracted.

When considering a mapping $f:V\left(G\right)\to V\left(H\right)$, the concept of f contracting an edge $\{x,y\}$ denotes that both vertices x and y are mapped to the same vertex in $V\left(H\right)$, i.e., $f\left(x\right)=f\left(y\right)$. The central concept is that homomorphisms must preserve the edges. If we also have the option to contract edges, then this achievement can be realized using regular homomorphisms when our graphs contain a loop at every vertex.

A mapping $f:V\left(G\right)\to V\left(H\right)$ is a weak homomorphism from a graph G to a graph H (also referred to as an egamorphism) if f contracts or preserves the edges, that is, $f\left(x\right)=f\left(y\right)$ or $\left\{f\right(x),f(y\left)\right\}\in E\left(H\right)$ whenever $\{x,y\}\in E\left(G\right)$. A weak homomorphism from G to itself is referred to as a weak endomorphism on G. We denote the set of weak homomorphisms from G to H as WHom($G,H$) and the set of weak endomorphisms on G as WEnd(G). It is evident that WEnd(G) constitutes a monoid under the composition of mappings. The composition of (weak) homomorphisms also forms a (weak) homomorphism. Consequently, this results in a preorder on graphs and defines a category [1].

Weak homomorphisms extend traditional graph mappings by allowing for edge contraction, offering a versatile tool for analyzing complex networks across disciplines. In computer science, they efficiently simplify large social networks by merging related users, revealing community structures. This ability to abstract while preserving key relationships makes weak homomorphisms valuable for diverse applications.

In 2010, Sirisathianwatthana and Pipattanajinda [9] established the count of weak homomorphisms of cycles as WHom($C_m,C_n$), expressed in terms of the collection of ${\mathrm{WHom}}_j^i$($P_{m-1},C_n$), where ${\mathrm{WHom}}_j^i$($P_{m-1},C_n$) represents a set of weak homomorphisms from $P_{m-1}$ to $C_n$, with the conditions that $f\left(0\right)=i$ and $f(m-1)=j$.

Recently, in 2022, Promsri et al. [10] introduced the number of weak homomorphisms of paths WHom($P_m,P_n$), by associating it with the order of the following three sets: $A_{m-1,n}^i=\{f\in$ WHom$(P_{m-1},P_n)|f\left(0\right)=i\}$, $B_{m-1,n}^i=\{f\in$ WHom$(P_{m-1},P_n)|f\left(0\right)=i\mathrm{and}f(m-2)=0\}$, and $C_{m-1,n}^i=\{f\in$ WHom$(P_{m-1},P_n)|f\left(0\right)=i\mathrm{and}f(m-2)=n-1\}$, where i ranges from 0 to $n-1$. This motivated us to investigate the number of weak homomorphisms from paths to the product of paths.

For any two graphs $G_1$ and $G_2$, the Cartesian product of $G_1$ and $G_2$ is the graph $G_1\square G_2$ with vertices $V(G_1\square G_2)=V\left(G_1\right)\times V\left(G_2\right)$, in which $\left\{\right(a,u),(b,v\left)\right\}$ forms an edge if either $a=b$ and $\{u,v\}\in E\left(G_2\right)$, or $\{a,b\}\in E\left(G_1\right)$ and $u=v$.

A rectangular grid graph $P_n\square P_k$ represents the Cartesian product of $P_n$ and $P_k$. These graphs, especially when $n=k$ (forming square grids, $P_n\square P_n$), possess a high degree of symmetry, including reflectional, rotational, and near-translational symmetry, a direct consequence of their regular structure. A key connection between paths and these grid graphs arises when considering the mappings between them. Specifically, a mapping f from the vertices of a path $P_m$ to the vertices of the Cartesian product graph $G_1\square G_2$ (which encompasses rectangular grids) is a homomorphism, if and only if, the sequence of vertices $f\left(0\right),f\left(1\right),\dots ,f(m-1)$ forms a walk within $G_1\square G_2$. This crucial observation establishes a one-to-one correspondence between the set of homomorphisms $f:P_m\to G_1\square G_2$ and the set of walks consisting of m vertices within $G_1\square G_2$. Similarly, we can establish a one-to-one correspondence between the set WHom($P_m,G_1\square G_2$) and the collection of partial walks with m vertices in $G_1\square G_2$. Here, the partial walk is a sequence obtained by concatenating q walks, namely $W_1,W_2,\dots ,W_q$, for some $q\in \mathbb{N}$, and the ending vertex of $W_i$ is the same as the starting vertex of $W_{i+1}$ for all $i=1,2,\dots ,q-1$.

Yingtaweesittikul et al. (2023) [11] successfully developed formulas for the number of weak homomorphisms from path graphs to ladder graphs ($P_n\square P_2$) and stacked prism graphs ($P_n\square C_k$). Here, we aim to determine formulas for the number of weak homomorphisms from the path graphs to rectangular grid graphs, denoted as |$\mathrm{WHom}(P_m,P_n\square P_k)$|, which provides a solution to the problem concerning the number of partial walks of m vertices within the rectangular grid graphs $P_n\square P_k$.

2. Basic Results and Examples

In 2018, Knauer and Pipattanajinda [12] introduced the count of weak endomorphisms on paths by relating it to the quantities of the shortest paths from the origin point $(0,0,0)$ to any arbitrary point $(i,j,k)$ within the three-dimensional square lattice, as well as within the r-ladder three-dimensional square lattice. Moreover, they provided formulas for the count of shortest paths from the point $(0,0,0)$ to any point $(i,j,k)$, as shown in Proposition 1. Figure 1 and Figure 2 depict the cubic lattice and the 2-ladder cubic lattice when $i=6$, $j=4$, and $k=4$, respectively.

Proposition 1

([12]). The numbers $M(i,j,k)$ and $M_r(i,j,k)$ of the shortest paths from the point $(0,0,0)$ to any point $(i,j,k)$ in the cubic lattice and in the r-ladder cubic lattice are

$$M(i,j,k)=\left({\displaystyle \genfrac{}{}{0pt}{}{i+j+k}{i,j,k}}\right)$$

and

$$M_r(i,j,k)=\left[\left({\displaystyle \genfrac{}{}{0pt}{}{i+j+k}{i,j,k}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{i+j+k}{j-r-1,i+r+1,k}}\right)\right],$$

respectively.

In 2023, Yingtaweesittikul et al. [13] introduced a formula to determine the count of homomorphisms from $P_m$ to $P_n\square P_k$, relating it to the order of the set of weak homomorphisms f from $P_m$ to $P_n$ with $f\left(0\right)=j$, denoted as ${\mathrm{Hom}}^j$($P_m,P_n$). This formula provides the solution to the problem concerning the number of walks of order m in the rectangular grid graphs $P_n\square P_k$. Moreover, they provided formulas for |${\mathrm{Hom}}^j$($P_m,P_n$)|, as shown in Theorem 1.

Theorem 1

([13]). Let $m,n$ be positive integers and j be a non-negative integer. Let $\mathcal{L}=max\{0,\lceil {\displaystyle \frac{m-j-1}{2}}\rceil \}$ and $\mathcal{U}=min\{m-1,\lfloor {\displaystyle \frac{m+n-j-2}{2}}\rfloor \}$. Then,

$$|{\mathrm{Hom}}^j(P_m,P_n)|=\sum _{i=\mathcal{L}}^{\mathcal{U}}\sum _{|t|\le \lfloor {\displaystyle \frac{m+n}{n}}\rfloor }\left(\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i-t(n+1)}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j-t(n+1)+1}}\right)\right).$$

(1)

If we let $m\le n$, let $j<n$, and reduce all of the zero terms, we can obtain the following corollary.

Corollary 1.

Let $m,n$ be positive integers and j be a non-negative integer such that $m\le n$ and $j<n$. Then,

$$\begin{array}{cc}\hfill |{\mathrm{Hom}}^j(P_m,P_n)|=& \sum _{t=max\left\{0,⌈{\displaystyle \frac{j-(n-m)}{2}}⌉\right\}}^{⌈{\displaystyle \frac{m+j}{2}}⌉-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right)-\sum _{t=0}^{⌊{\displaystyle \frac{j-(n-m)}{2}}⌋-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right)-\sum _{t=0}^{⌊{\displaystyle \frac{m-j-1}{2}}⌋-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right).\hfill \end{array}$$

Proof. 

As $m\le n$, $\lfloor {\displaystyle \frac{m+n}{n}}\rfloor \le 2$. Thus, $t\in \{-2,-1,0,1,2\}$ and Equation (1) can be reduced to

$$\begin{array}{cc}\hfill |{\mathrm{Hom}}^j(P_m,P_n)|=& \sum _{i=\mathcal{L}}^{\mathcal{U}}\left(\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+2n+2}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j+2n+3}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+n+1}}\right)\right.\hfill \\ & -\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j+n+2}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j+1}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i-n-1}}\right)\hfill \\ & \left.-\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j-n}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i-2n-2}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j-2n-1}}\right)\right).\hfill \end{array}$$

As $\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+2n+2}}\right)$, $\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j+2n+3}}\right)$, $\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+n+1}}\right)$, $\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j+n+2}}\right)$, $\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i-n-1}}\right)$, $\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i-2n-2}}\right)$, and $\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j-2n-1}}\right)$ are all zeros, we have

$$\begin{array}{cc}\hfill |{\mathrm{Hom}}^j(P_m,P_n)|& =\sum _{i=\mathcal{L}}^{\mathcal{U}}\left(\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j+1}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j-n}}\right)\right)\hfill \\ \hfill & =\sum _{i=\mathcal{L}}^{\mathcal{U}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i}}\right)-\sum _{i=\mathcal{L}}^{\mathcal{U}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j+1}}\right)-\sum _{i=\mathcal{L}}^{\mathcal{U}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j-n}}\right)\hfill \\ \hfill & =\sum _{i=\mathcal{L}}^{\mathcal{U}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{(m-1)-i}}\right)-\sum _{i=\mathcal{L}}^{\mathcal{U}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{(m-1)-i-j-1}}\right)-\sum _{i=\mathcal{L}}^{\mathcal{U}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{i+j-n}}\right)\hfill \\ \hfill & =\sum _{t=(m-1)-\mathcal{U}}^{(m-1)-\mathcal{L}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right)-\sum _{t=(m-1)-j-1-\mathcal{U}}^{(m-1)-j-1-\mathcal{L}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right)-\sum _{t=\mathcal{L}+j-n}^{\mathcal{U}+j-n}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right)\hfill \\ \hfill & =\sum _{t=max\left\{0,⌈{\displaystyle \frac{j-(n-m)}{2}}⌉\right\}}^{⌈{\displaystyle \frac{m+j}{2}}⌉-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right)-\sum _{t=0}^{⌊{\displaystyle \frac{m-j-1}{2}}⌋-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right)-\sum _{t=0}^{⌊{\displaystyle \frac{j-(n-m)}{2}}⌋-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{t}}\right).\hfill \end{array}$$

□

To better understand the main theorem, we start by examining a straightforward example. Our goal at this stage is to create a visual representation of weak homomorphisms. Check Figure 3 for potential weak homomorphisms from $P_4$ to $P_5$ specifically mapping 0 to 0. The numbers at the top represent elements of the domain set $V\left(P_4\right)$, and those on the left correspond to elements of the image set $V\left(P_5\right)$.

The mappings $f_1,f_2\in {\mathrm{WHom}}^0(P_4,P_5)$ with $f_1\left(0\right)=0,f_1\left(1\right)=0,f_1\left(2\right)=0,f_1\left(3\right)=0$ and $f_2\left(0\right)=0,f_2\left(1\right)=1,f_2\left(2\right)=2,f_2\left(3\right)=3$ are represented by the dotted line on the top and black line, respectively (see Figure 4).

Figure 5 illustrates weak homomorphisms using the cubic lattice. Multiple cases need consideration. Initially, when $f(x+1)=f\left(x\right)+1$, it corresponds to moving from $(i,j,k)$ to $(i+1,j,k)$. Similarly, when $f(x+1)=f\left(x\right)-1$, it corresponds to moving from $(i,j,k)$ to $(i,j+1,k)$. For the remaining cases, where $f(x+1)=f\left(x\right)$, the correspondence involves moving from $(i,j,k)$ to $(i,j,k+1)$. Consequently, the mappings $f_1$ and $f_2$ are depicted by the shortest paths from $(0,0,0)$ to $(0,0,3)$ and $(3,0,0)$ in the 0-ladder cubic lattice, respectively (see Figure 6).

The cardinality $|{\mathrm{WHom}}^0(P_4,P_5)|$ is the summation of $M(i,j,k)$ and $M_0(i,j,k)$ where $i+j+k=3$ (large black points). From Figure 5, if $j\le 0$, we use $M(i,j,k)$; otherwise, $M_0(i,j,k)$.

$$\begin{array}{cc}\hfill |{\mathrm{WHom}}^0(P_4,P_5)|& =M(3,0,0)+M_0(2,1,0)+M(2,0,1)+M_0(1,1,1)\hfill \\ \hfill & +M(1,0,2)+M(0,0,3)\hfill \\ \hfill & =\left({\displaystyle \genfrac{}{}{0pt}{}{3}{3,0,0}}\right)+\left[\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2,1,0}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0,3,0}}\right)\right]+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2,0,1}}\right)\hfill \\ \hfill & +\left[\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1,1,1}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0,2,1}}\right)\right]+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1,0,2}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0,0,3}}\right)\hfill \\ \hfill & =13.\hfill \end{array}$$

Similar to the above example, Figure 7 visualizes the possible weak homomorphisms of the path $P_4$ to $P_5$ which map 0 to 1.

The cardinality $|{\mathrm{WHom}}^1(P_4,P_5)|$ is the summation of $M(i,j,k)$ and $M_1(i,j,k)$ where $i+j+k=3$ (large black points). From Figure 8, if $j\le 1$, we use $M(i,j,k)$; otherwise $M_1(i,j,k)$.

$$\begin{array}{cc}\hfill |{\mathrm{WHom}}^1(P_4,& P_5\left)\right|=M(2,1,0)+M_1(1,2,0)+M(2,0,1)+M(1,1,1)\hfill \\ \hfill & +M(1,0,2)+M(0,1,2)+M(0,0,3)+M(3,0,0)\hfill \\ \hfill & =\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2,1,0}}\right)+\left[\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1,2,0}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0,3,0}}\right)\right]+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2,0,1}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1,1,1}}\right)\hfill \\ \hfill & +\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1,0,2}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0,1,2}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0,0,3}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{3,0,0}}\right)\hfill \\ \hfill & =22.\hfill \end{array}$$

3. The Number of (Weak) Homomorphisms from Paths to Paths That Map 0 to $\mathit{j}$

In this section, we present the formula for determining the count of weak homomorphisms from paths $P_m$ to $P_n$, where 0 is mapped to j. We represent the set of weak homomorphisms from $P_m$ to $P_n$, with the mapping of 0 to j, as ${\mathrm{WHom}}^j$($P_m,P_n$).

Theorem 2.

Let $m,n$ be positive integers and j be a non-negative integer, such that $m\le n$ and $j<n$. Then,

$$\begin{array}{cc}\hfill |{\mathrm{WHom}}^j& (P_m,P_n)|=\sum _{t=j+1}^{j+⌊{\displaystyle \frac{m-j-1}{2}}⌋}\sum _{s=t-j}^{m-1-t}\left[\left(\genfrac{}{}{0pt}{}{m-1}{s,t,m-1-s-t}\right)-\left(\genfrac{}{}{0pt}{}{m-1}{t-j-1,s+j+1,m-1-s-t}\right)\right]\hfill \\ \hfill & +\sum _{t=max\{j-n+m+1,0\}}^j\sum _{s=0}^{m-1-t}\left(\genfrac{}{}{0pt}{}{m-1}{s,t,m-1-s-t}\right)+\sum _{t=0}^{j-n+m}\sum _{s=0}^{n-j-1}\left(\genfrac{}{}{0pt}{}{m-1}{s,t,m-1-s-t}\right)\hfill \\ \hfill & +\sum _{t=n-j-1+1}^{n-j-1+⌊{\displaystyle \frac{j-n+m}{2}}⌋}\sum _{s=t-(n-j-1)}^{m-1-t}\left[\left(\genfrac{}{}{0pt}{}{m-1}{s,t,m-1-s-t}\right)-\left(\genfrac{}{}{0pt}{}{m-1}{t-n+j,s+n-j,m-1-s-t}\right)\right].\hfill \end{array}$$

Proof. 

To find $|{\mathrm{WHom}}^j(P_m,P_n)|$, we count the number of shortest paths from the point $(0,0,0)$ to any point $(i_0,j_0,k_0)$, where $i_0+j_0+k_0=m-1$ in the j-ladder cubic lattice. Consider the following four different cases corresponding to the value of $j_0$. For each case, the points $(i_0,j_0)$ are on the blue arrows in Figure 9, Figure 10, Figure 11 and Figure 12, respectively.

• Case 1: $j_0>j$. For each $j_0=j+t$, there are ${\sum }_{i_0=t}^{m-j-1-t}{M}_j(i_0,j_0,k_0)$ shortest paths.
  As $t\le {\displaystyle \frac{m-j-1}{2}}$, we obtain

  $$\begin{array}{cc}\hfill \sum _{t=1}^{⌊{\displaystyle \frac{m-j-1}{2}}⌋}\sum _{i_0=t}^{m-j-1-t}& M_j(i_0,j+t,k_0)=\sum _{t=j+1}^{j+⌊{\displaystyle \frac{m-j-1}{2}}⌋}\sum _{s=t-j}^{m-1-t}{M}_j(s,t,m-1-s-t)\hfill \\ \hfill & =\sum _{t=j+1}^{j+⌊{\displaystyle \frac{m-j-1}{2}}⌋}\sum _{s=t-j}^{m-1-t}\left[\left(\genfrac{}{}{0pt}{}{m-1}{s,t,m-1-s-t}\right)-\left(\genfrac{}{}{0pt}{}{m-1}{t-j-1,s+j+1,m-1-s-t}\right)\right].\hfill \end{array}$$
• Case 2: $j-n+m<j_0\le j$ and $i_0<n-j-1$. For each $j_0=j-t$, there are ${\sum }_{i_0=0}^{m-j-1+t}M(i_0,j_0,k_0)$ shortest paths. Since $t<n-m$, we obtain

  $$\begin{array}{cc}\hfill \sum _{t=0}^{n-m-1}\sum _{i_0=0}^{m-j-1+t}M(i_0,j-t,k_0)& =\sum _{t=max\{j-n+m+1,0\}}^j\sum _{s=0}^{m-1-t}M(s,t,m-1-s-t)\hfill \\ \hfill & =\sum _{t=max\{j-n+m+1,0\}}^j\sum _{s=0}^{m-1-t}\left(\genfrac{}{}{0pt}{}{m-1}{s,t,m-1-s-t}\right).\hfill \end{array}$$
• Case 3: $j_0\le j-n+m\le j$ and $i_0\le n-j-1$. For each $i_0,j_0$, there are $M(i_0,j_0,k_0)$ shortest paths. We obtain

  $$\begin{array}{cc}\hfill \sum _{j_0=0}^{j-n+m}\sum _{i_0=0}^{n-j-1}M(i_0,j_0,k_0)& =\sum _{t=0}^{j-n+m}\sum _{s=0}^{n-j-1}M(s,t,m-1-s-t)\hfill \\ \hfill & =\sum _{t=0}^{j-n+m}\sum _{s=0}^{n-j-1}\left(\genfrac{}{}{0pt}{}{m-1}{s,t,m-1-s-t}\right).\hfill \end{array}$$
• Case 4: $j_0\le j$ and $i_0>n-j-1$. For each $i_0=n-j-1+t$, there are ${\sum }_{j_0=t}^{j-n+m-t}{M}_{n-j-1}(j_0,i_0,k_0)$ shortest paths. This can be obtained by flipping the cubic lattice diagonally.

As $t\le {\displaystyle \frac{j-n+m}{2}}$, we obtain

$$\begin{array}{cc}\hfill \sum _{t=1}^{⌊{\displaystyle \frac{j-n+m}{2}}⌋}& \sum _{j_0=t}^{j-n+m-t}{M}_{n-j-1}(j_0,n-j-1+t,k_0)\hfill \\ & =\sum _{t=n-j-1+1}^{n-j-1+⌊{\displaystyle \frac{j-n+m}{2}}⌋}\sum _{s=t-(n-j-1)}^{m-1-t}{M}_{n-j-1}(s,t,m-1-s-t)\hfill \\ \hfill & =\sum _{t=n-j-1+1}^{n-j-1+⌊{\displaystyle \frac{j-n+m}{2}}⌋}\sum _{s=t-(n-j-1)}^{m-1-t}\left[\left(\genfrac{}{}{0pt}{}{m-1}{s,t,m-1-s-t}\right)-\left(\genfrac{}{}{0pt}{}{m-1}{t-n+j,s+n-j,m-1-s-t}\right)\right].\hfill \end{array}$$

Adding up over all of the cases, $|{\mathrm{WHom}}^j(P_m,P_n)|$ is as desired. □

For convenience, we compute $|{\mathrm{WHom}}^j(P_m,P_n)|$ and $|{\mathrm{Hom}}^j(P_m,P_n)|$ for $2\le m\le n\le 8$. The results are presented in

Table 1. Numbers of weak homomorphisms $f:P_m\to P_n$ where $f\left(0\right)=j$ for $2\le m\le n\le 8.$

		n
$\mathit{m}$	$\mathit{j}$	2	3	4	5	6	7	8
2	0	2	2	2	2	2	2	2
	1		3	3	3	3	3	3
	2				3	3	3	3
	3						3	3
3	0		5	5	5	5	5	5
	1		7	8	8	8	8	8
	2				9	9	9	9
	3						9	9
4	0			13	13	13	13	13
	1			21	22	22	22	22
	2				25	26	26	26
	3						27	27
5	0				35	35	35	35
	1				60	61	61	61
	2				69	74	75	75
	3						79	80
6	0					96	96	96
	1					170	171	171
	2					209	215	216
	3						229	235
7	0						267	267
	1						482	483
	2						615	622
	3						659	686
8	0							750
	1							1372
	2							1791
	3							1994

and

Table 2. Numbers of homomorphisms $f:P_m\to P_n$ where $f\left(0\right)=j$ for $2\le m\le n\le 8.$

		n
$\mathit{m}$	$\mathit{j}$	2	3	4	5	6	7	8
2	0	1	1	1	1	1	1	1
	1		2	2	2	2	2	2
	2				2	2	2	2
	3						2	2
3	0		2	2	2	2	2	2
	1		2	3	3	3	3	3
	2				4	4	4	4
	3						4	4
4	0			3	3	3	3	3
	1			5	6	6	6	6
	2				6	7	7	7
	3						8	8
5	0				6	6	6	6
	1				9	10	10	10
	2				12	13	14	14
	3						14	15
6	0					10	10	10
	1					19	20	20
	2					23	24	25
	3						28	29
7	0						20	20
	1						34	35
	2						48	49
	3						48	54
8	0							35
	1							69
	2							89
	3							103

, respectively.

4. The Number of Weak Homomorphisms from Paths to Grid Graphs

In this section, we present the formulas for determining the count of weak homomorphisms from paths $P_m$ to rectangular grid graphs $P_n\square P_k$. We represent the set of weak homomorphisms from $P_m$ to $P_n\square P_k$, mapping 0 to $(i,j)$, as ${\mathrm{WHom}}^{ij}(P_m,P_n\square P_k)$. From the symmetry of $P_n\square P_k$, we deduce the following lemma:

Lemma 1.

Let i and n be integers, such that $0\le j<n$, and let $m>2$ be a positive integer.

1. 

$|{\mathrm{WHom}}^{ij}(P_m,P_n\square P_k)|=|{\mathrm{WHom}}^{(n-i-1)j}(P_m,P_n\square P_k)|$

$=|{\mathrm{WHom}}^{i(k-j-1)}(P_m,P_n\square P_k)|$

$=|{\mathrm{WHom}}^{(n-i-1)(k-j-1)}(P_m,P_n\square P_k)|$,

    for all $i\in \{0,1,\dots ,n-1\}$ and $j\in \{0,1,\cdots ,k-1\}$.

2. 

$|\mathrm{WHom}(P_m,P_{2n}\square P_{2k})|=4{\sum }_{i=0}^{n-1}{\sum }_{j=0}^{k-1}\left|{\mathrm{WHom}}^{ij}(P_m,P_{2n}\square P_{2k})\right|$.

3. 

$|\mathrm{WHom}(P_m,P_{2n+1}\square P_{2k})|=4{\sum }_{i=0}^{n-1}{\sum }_{j=0}^{k-1}\left|{\mathrm{WHom}}^{ij}(P_m,P_{2n+1}\square P_{2k})\right|$

$+2{\sum }_{j=0}^{k-1}\left|{\mathrm{WHom}}^{nj}(P_m,P_{2n+1}\square P_{2k})\right|$.

4. 

$|\mathrm{WHom}(P_m,P_{2n}\square P_{2k+1})|=4{\sum }_{i=0}^{n-1}{\sum }_{j=0}^{k-1}\left|{\mathrm{WHom}}^{ij}(P_m,P_{2n}\square P_{2k+1})\right|$

$+2{\sum }_{i=0}^{n-1}\left|{\mathrm{WHom}}^{ik}(P_m,P_{2n}\square P_{2k+1})\right|$.

5. 

$\left|\mathrm{WHom}\right(P_m,P_{2n+1}\square P_{2k+1}\left)\right|$

$=4{\sum }_{i=0}^{n-1}{\sum }_{j=0}^{k-1}\left|{\mathrm{WHom}}^{ij}(P_m,P_{2n+1}\square P_{2k+1})\right|$

$+2{\sum }_{j=0}^{k-1}\left|{\mathrm{WHom}}^{nj}(P_m,P_{2n+1}\square P_{2k+1})\right|$

$+2{\sum }_{i=0}^{n-1}\left|{\mathrm{WHom}}^{ik}(P_m,P_{2n+1}\square P_{2k+1})\right|$

$+|{\mathrm{WHom}}^{nk}(P_m,P_{2n+1}\square P_{2k+1})|$.

Example 1.

$|{\mathrm{WHom}}^{00}(P_4,P_4\square P_5)|=43$.

Figure 13 shows all possible weak homomorphisms from $P_4$ to $P_4\square P_5$ which map 0 to $(0,0)$. The numbers on top are the elements of domain set $V\left(P_4\right)$ and the tuples on the left are elements of image set $V(P_4\square P_5)$. The tuples with the same second elements are represented by the circle with the same color.

We noted that normal black lines represent the increment of the first coordinate, dashed black lines represent the decrement of the first coordinate, normal magenta lines represent the increment of the second coordinate, magenta lines represent the decrement of the second coordinate, and cyan lines represent no change in both coordinates.

We now divide all the mappings in ${\mathrm{WHom}}^{00}(P_4,P_4\square P_5)$ into groups according to the number of changes in the first coordinate h, and rewrite each path as two shorter paths (see

Table 3. The mappings in ${\mathrm{WHom}}^{00}(P_4,P_4\square P_5)$ grouped by changes in the first coordinate (h). Each path is divided into two: gray lines (first path) and cyan/magenta lines (second path).

h	$\mathit{f}\in {\mathbf{WHom}}^{00}({\mathit{P}}_4,{\mathit{P}}_4\square {\mathit{P}}_5)$ with Changes in the First Coordinate h Times	Paths Represent Each $\mathit{f}\in {\mathbf{WHom}}^{00}({\mathit{P}}_4,{\mathit{P}}_4\square {\mathit{P}}_5)$ (Expanded Diagram)	${\mathit{P}}_{\mathit{h}+1}\mathbf{and}{\mathit{P}}_{4-\mathit{h}}$
0
1
2
3

). The first path is formed by gray lines. On the other hand, the second path consists of cyan and magenta lines. In both paths, the lines are arranged in sequential order.

$$\begin{array}{cc}\hfill |{\mathrm{WHom}}^{00}(P_4,P_4\square P_5)|& =\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0}}\right)|{\mathrm{Hom}}^0(P_1,P_4)||{\mathrm{WHom}}^0(P_4,P_5)|\hfill \\ & +\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1}}\right)|{\mathrm{Hom}}^0(P_2,P_4)||{\mathrm{WHom}}^0(P_3,P_5)|\hfill \\ \hfill & +\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2}}\right)|{\mathrm{Hom}}^0(P_3,P_4)||{\mathrm{WHom}}^0(P_2,P_5)|\hfill \\ & +\left({\displaystyle \genfrac{}{}{0pt}{}{3}{3}}\right)|{\mathrm{Hom}}^0(P_4,P_4)||{\mathrm{WHom}}^0(P_1,P_5)|\hfill \\ \hfill & =1\left(1\right)\left(13\right)+3\left(1\right)\left(5\right)+3\left(2\right)\left(2\right)+1\left(3\right)\left(1\right)\hfill \\ \hfill & =43.\hfill \end{array}$$

Theorem 3.

Let $m,n$ and k be positive integers and $i,j$ be non-negative integers, such that $i<{\displaystyle \frac{n}{2}}-1$ and $j<{\displaystyle \frac{k}{2}}-1$. It follows that

$$|{\mathrm{WHom}}^{ij}(P_m,P_n\square P_k)|=\sum _{h=0}^{m-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{h}}\right)|{\mathrm{Hom}}^i(P_{h+1},P_n)||{\mathrm{WHom}}^j(P_{m-h},P_k)|.$$

Proof. 

Let $f\in {\mathrm{WHom}}^{ij}(P_m,P_n\square P_k)$. For each $x\in \{0,1,m-2\}$ in the domain, either $f(x+1)=f\left(x\right)\pm (1,0)$ or $f(x+1)=f\left(x\right)\pm (0,t)$, where $t\in \{0,1\}$. Assume changes in the first coordinate appear h times. Then, changes in the second coordinate appear $m-1-h$ times. The sequence of changes in the first coordinate form a homomorphism $f_1\in {\mathrm{Hom}}^i(P_{h+1},P_n)$. Similarly, the sequence of remaining changes (and no changes) in the second coordinate form a weak homomorphism $f_2\in {\mathrm{WHom}}^i(P_{m-1-h+1},P_k)$. Thus, the corresponding path graph of f can be obtained from the permutations of all edges in path graphs of $f_1$ and $f_2$ with a fixed sequential order. There are $\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{h}}\right)$ permutations in total. Hence, $|{\mathrm{WHom}}^{ij}(P_m,P_n\square P_k)|={\sum }_{h=0}^{m-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{h}}\right)|{\mathrm{Hom}}^i(P_{h+1},P_n)||{\mathrm{WHom}}^j(P_{m-h},P_k)|.$ □

From Lemma 1 and Theorem 3, we obtain the theorem below.

Theorem 4.

The cardinalities $\left|\mathrm{WHom}\right(P_m,P_n\square P_k\left)\right|$ of the weak homomorphisms from undirected paths $P_m$ to grid graphs $P_n\square P_k$ are

$$\begin{array}{cc}|\mathrm{WHom}(P_m,P_n\square P_k)|=& 4{\sum }_{i=0}^{\lfloor n/2\rfloor -1}{\sum }_{j=0}^{\lfloor k/2\rfloor -1}\left|{\mathrm{WHom}}^{ij}(P_m,P_n\square P_k)\right|\hfill \\ & +(1-{(-1)}^n){\sum }_{j=0}^{\lfloor k/2\rfloor -1}\left|{\mathrm{WHom}}^{\lfloor n/2\rfloor j}(P_m,P_n\square P_k)\right|\hfill \\ & +(1-{(-1)}^k){\sum }_{i=0}^{\lfloor n/2\rfloor -1}\left|{\mathrm{WHom}}^{i\lfloor k/2\rfloor }(P_m,P_n\square P_k)\right|\hfill \\ & +(1/4)(1-{(-1)}^n)(1-{(-1)}^k)\left|{\mathrm{WHom}}^{\lfloor n/2\rfloor \lfloor k/2\rfloor }(P_m,P_n\square P_k)\right|\hfill \end{array}$$

where $|{\mathrm{WHom}}^{ij}(P_m,P_n\square P_k)|={\sum }_{h=0}^{m-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{h}}\right)|{\mathrm{Hom}}^i(P_{h+1},P_n)||{\mathrm{WHom}}^j(P_{m-h},P_k)|$.

For convenience, we compute $\left|\mathrm{WHom}\right(P_m,P_n\square P_k\left)\right|$ for $2\le m\le n,k\le 8.$ The results are presented in

Table 4. Numbers of weak homomorphisms $f:P_m\to P_n\square P_k$ for $2\le m\le n,k\le 8.$

		k
$\mathit{m}$	$\mathit{n}$	2	3	4	5	6	7	8
2	2	12	20	28	36	44	52	60
	3	20	33	46	59	72	85	98
	4	28	46	64	82	100	118	136
	5	36	59	82	105	128	151	174
	6	44	72	100	128	156	184	212
	7	52	85	118	151	184	217	250
	8	60	98	136	174	212	250	288
3	3		125	182	239	296	353	410
	4		182	264	346	428	510	592
	5		239	346	453	560	667	774
	6		296	428	560	692	824	956
	7		353	510	667	824	981	1138
	8		410	592	774	956	1138	1320
4	4			1104	1480	1856	2232	2608
	5			1480	1981	2482	2983	3484
	6			1856	2482	3108	3734	4360
	7			2232	2983	3734	4485	5236
	8			2608	3484	4360	5236	6112
5	5				8733	11,088	13,443	15,798
	6				11,088	14,068	17,048	20,028
	7				13,443	17,048	20,653	24,258
	8				15,798	20,028	24,258	28,488
6	6					64,004	78,226	92,448
	7					78,226	95,573	112,920
	8					92,448	112,920	133,392
7	7						443,833	527,452
	8						527,452	626,696
8	8							2,951,832

.

5. Applications of the Number of Weak Homomorphisms from Paths to Rectangular Grid Graphs

The concept of partial walks, defined as sequences of concatenated walks, where the end of one walk coincides with the start of the next, offers a powerful tool for modeling and analyzing diverse phenomena within rectangular grid graphs. Crucially, the number of such partial walks with length m within a rectangular grid graph $P_n\square P_k$ directly captured by the number of weak homomorphisms from a path graph $P_m$ to $P_n\square P_k$. This connection provides a powerful mathematical framework for enumerating and analyzing these walks. This approach finds particular relevance in scenarios involving segmented or interrupted movement, information flow, and pattern recognition.

In robotics and path planning, the enumeration of these partial walks can represent the count of possible robot navigation paths through complex environments, accommodating pauses, task-specific stops, and dynamic obstacle avoidance. For instance, knowing this count allows us to quantify the number of distinct m-step movement sequences a robot could take. In network analysis and data routing, this enumeration models the count of possible packet transmission paths with delays or interruptions, effectively capturing the flow of information through networks where nodes introduce pauses or processing steps. Similarly, in image processing and computer vision, this enumeration quantifies the number of ways to trace pixel connectivity in edge detection or pattern recognition, allowing for the statistical analysis of patterns composed of non-contiguous segments. Moreover, in combinatorial problems and game theory, this enumeration enables the modeling and counting of constrained movements, such as player actions in grid-based games, and provides a framework for enumerating the number of possible sequences of actions in scenarios with intermittent actions. By leveraging the connection between the enumeration of weak homomorphisms and the counting of partial walks, we gain a deeper quantitative understanding of a wide range of real-world phenomena represented by rectangular grid graphs.

6. Conclusions

This paper has introduced formulas aimed at determining the number of weak homomorphisms from paths $P_m$ to rectangular grid graphs $P_n\square P_k$. These formulas are expressed in terms of $|{\mathrm{Hom}}^i(P_{h+1},P_n)|$ and $|{\mathrm{WHom}}^j(P_{m-h},P_k)|$ for all $h=0,1,\dots ,m-1$. The proposed formulas serve as a solution to the specific problem of enumerating the number of partial walks of m vertices within the rectangular grid graphs.