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Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).

Given a pair of non-negative integers and , (,) denotes a square lattice graph with a vertex set {0,1,2,..., – 1} × {0,1,2,..., – 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph (,) has a vertex set {( + ) |  ∈ {0,1,2,..., − 1},  ∈ {0,1,2,..., − 1}} where , and an edge set consists of a pair of vertices with unit distance. Let (,) and (,) be the th power of the graph (,) and (,), respectively. Given an undirected graph = (,) and a non-negative vertex weight function , a multicoloring of is an assignment of colors to such that each vertex ∈ admits () colors and every adjacent pair of two vertices does not share a common color.

In this paper, we show necessary and sufficient conditions that [∀ , (,) is perfect] and/or [∀ , (,) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring ((,),) and ((,),).