Linear Coloring Research Articles

Linear list coloring of some sparse graph

Linear list coloring of some sparse graph

On linear coloring of planar graphs with small girth

A vertex coloring of a graph G is linear if the subgraph induced by the vertices of any two color classes is the union of vertex-disjoint paths. In this paper, we study the linear coloring of graphs with small girth and prove that: (1) Every planar graph with maximum degree Δ≥39 and girth g≥6 is linearly (⌈Δ2⌉+1)-colorable. (2) There exists an integer Δ0 such that every planar graph with maximum degree Δ≥Δ0 and girth g≥5 is linearly (⌈Δ2⌉+1)-colorable. The latter result is best possible in some sense.

Linear colorings of subcubic graphs

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induces a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is C5 or K3,3. This confirms a conjecture raised by Esperet, Montassier and Raspaud [L. Esperet, M. Montassier, and A. Raspaud, Linear choosability of graphs, Discrete Math. 308 (2008) 3938–3950]. Our proof is constructive and yields a linear-time algorithm to find such a coloring.

A result on linear coloring of planar graphs

A proper vertex coloring of a graph is called linear if the subgraph induced by the vertices colored by any two colors is a set of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by lc(G), is the minimum number of colors in a linear coloring of G. In this paper, we show lc(G)⩽Δ(G)+7 for a planar graph G with maximum degree Δ(G).

Linear Coloring of Planar Graphs Without 4-Cycles

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc( G ) of G is the ...

Scratchpad memory allocation for arrays in permutation graphs

In modern embedded systems, on-chip memory is generally organized as software-managed scratchpad memory (SPM). Li et al. discovered that in many embedded applications, the array interference graphs tend to be superperfect graphs. Based on this, they presented a superperfect graph-based SPM allocation algorithm, which is the best in the literature. In this paper, we make further progress in proving that the array interference graphs they focused on are permutation graphs; that is, a subclass of a superperfect graph. Compared to the latter, the permutation graph has certain advantages, such as linear time recognition and linear time optimal interval coloring. Based on our theoretical results, we improve their algorithm by transforming it into a permutation graph-based one by means of simple plug-in revisions. The experiments confirm that the improved algorithm achieves optimal coloring, and therefore, better allocation than the original superperfect graph-based algorithm for a broader scope of array interference graphs.

Formula omitted]-forested choosability of planar graphs and sparse graphs

A proper vertex coloring of a simple graph G is k -forested if the subgraph induced by the vertices of any two color classes is a k -forest, i.e. a forest with maximum degree at most k . The 2 -forested coloring is also known as linear coloring, which has been extensively studied in the literature. In this paper, we aim to extend the study of 2 -forested coloring to general k -forested colorings for every k ≥ 3 by combinatorial means. Precisely, we prove that for a fixed integer k ≥ 3 and a planar graph G with maximum degree Δ ( G ) ≥ Δ and girth g ( G ) ≥ g , if ( Δ , g ) ∈ { ( k + 1 , 10 ) , ( 2 k + 1 , 8 ) , ( 4 k + 1 , 7 ) } , then the k -forested chromatic number of G is actually ⌈ Δ ( G ) k ⌉ + 1 . Moreover, we also prove that the k -forested chromatic number of a planar graph G with maximum degree Δ ( G ) ≥ k + 1 and girth g ( G ) ≥ 8 is ⌈ Δ ( G ) k ⌉ + 1 provided k ≥ 7 . In addition, we show that the k -forested chromatic number of an outerplanar graph G is at most ⌈ Δ ( G ) k ⌉ + 2 for every k ≥ 2 . In fact, all these results are proved for not only planar graphs but also for sparse graphs, i.e. graphs having a low maximum average degree mad ( G ) , and we actually prove a choosability version of these results.

Linear coloring of sparse graphs

A lineark-coloring of a graph G is a proper k-coloring of G such that the graph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic numberlc(G) of a graph G is the smallest number k such that G has a linear k-coloring. In this paper, we prove that if G is a graph with maximum degree Δ and maximum average degree mad(G)<145, then lc(G)≤⌈Δ2⌉+2. In particular, lc(G)≤⌈Δ2⌉+2 if G is a planar graph with maximum degree Δ and girth g(G)≥7. This improves some known results in this direction, and further supports a conjecture recently proposed by D. Cranston and G. Yu, which states that lc(G)≤⌈Δ2⌉+2 for every graph G with mad(G)<3.

New upper bounds on linear coloring of planar graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has (1) lc(G) ≤ Δ + 21 if Δ ≥ 9; (2) \(lc(G) \leqslant \left\lceil {\tfrac{\Delta } {2}} \right\rceil + 7\) if g ≥ 5; (3) \(lc(G) \leqslant \left\lceil {\tfrac{\Delta } {2}} \right\rceil + 2\) if g ≥ 7 and Δ≥ 7.

Linear choosability of sparse graphs

A linear coloring is a proper coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list chromatic number, denoted lc ℓ ( G ) , of sparse graphs. The maximum average degree of a graph G , denoted m a d ( G ) , is the maximum of the average degrees of all subgraphs of G . It is clear that any graph G with maximum degree Δ ( G ) satisfies lc ℓ ( G ) ≥ ⌈ Δ ( G ) / 2 ⌉ + 1 . In this paper, we prove the following results: (1) if mad ( G ) < 12 / 5 and Δ ( G ) ≥ 3 , then lc ℓ ( G ) = ⌈ Δ ( G ) / 2 ⌉ + 1 , and we give an infinite family of examples to show that this result is best possible; (2) if mad ( G ) < 3 and Δ ( G ) ≥ 9 , then lc ℓ ( G ) ≤ ⌈ Δ ( G ) / 2 ⌉ + 2 , and we give an infinite family of examples to show that the bound on mad ( G ) cannot be increased in general; (3) if G is planar and has girth at least 5, then lc ℓ ( G ) ≤ ⌈ Δ ( G ) / 2 ⌉ + 4 .

A Lebesgue's type theorem on toroidal graphs and its application to linear coloring

A coloring of a graph <italic>G</italic> is said to be linear if any two color classes induce an acyclic subgraph of maximum degree at most 2. In this paper, we first prove a Lebesgue's type theorem concerning the structure of toroidal graphs. As a consequence of this result, we show that every toroidal graph <italic>G</italic> of girth at least 5 can be 「(△(<italic>G</italic>))/2」+ 4)-linear list colorable, unless △(<italic>G</italic>) = 4 and G has a subgraph containing only faces of degree five of which each is incident with three vertices of degree 3 and two vertices of degree 4. This generalizes and improves some known results.

Upper bounds on the linear chromatic number of a graph

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic numberlc(G) of G is the smallest number of colors in a linear coloring of G.Let G be a graph with maximum degree Δ(G). In this paper we prove the following results: (1) lc(G)≤12(Δ(G)2+Δ(G)); (2) lc(G)≤8 if Δ(G)≤4; (3) lc(G)≤14 if Δ(G)≤5; (4) lc(G)≤⌊0.9Δ(G)⌋+5 if G is planar and Δ(G)≥52.

Improved bounds on linear coloring of plane graphs

Improved bounds on linear coloring of plane graphs

Improved bounds on linear coloring of plane graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we give some upper bounds on linear chromatic number for plane graphs with respect to their girth, that improve some results of Raspaud and Wang (2009).

Linear coloring of graphs embeddable in a surface of nonnegative characteristic

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G.

Linear coloring of planar graphs with large girth

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc ( G ) of the graph G is the smallest number of colors in a linear coloring of G . In this paper we prove that every planar graph G with girth g and maximum degree Δ has lc ( G ) = ⌈ Δ 2 ⌉ + 1 if G satisfies one of the following four conditions: (1) g ≥ 13 and Δ ≥ 3 ; (2) g ≥ 11 and Δ ≥ 5 ; (3) g ≥ 9 and Δ ≥ 7 ; (4) g ≥ 7 and Δ ≥ 13 . Moreover, we give better upper bounds of linear chromatic number for planar graphs with girth 5 or 6.

Linear colorings of simplicial complexes and collapsing

A vertex coloring of a simplicial complex Δ is called a linear coloring if it satisfies the property that for every pair of facets ( F 1 , F 2 ) of Δ, there exists no pair of vertices ( v 1 , v 2 ) with the same color such that v 1 ∈ F 1 ∖ F 2 and v 2 ∈ F 2 ∖ F 1 . The linear chromatic number lchr ( Δ ) of Δ is defined as the minimum integer k such that Δ has a linear coloring with k colors. We show that if Δ is a simplicial complex with lchr ( Δ ) = k , then it has a subcomplex Δ ′ with k vertices such that Δ is simple homotopy equivalent to Δ ′ . As a corollary, we obtain that lchr ( Δ ) ⩾ Homdim ( Δ ) + 2 . We also show in the case of linearly colored simplicial complexes, the usual assignment of a simplicial complex to a multicomplex has an inverse. Finally, we show that the chromatic number of a simple graph is bounded from above by the linear chromatic number of its neighborhood complex.

Routing and wavelength assignment in optical WDM networks with maximum quantity of edge disjoint paths

In the present paper routing and wavelength assignment (RWA) in optical WDM networks is discussed. Previous techniques on based on the integer linear programming and graph coloring are complex and require extensive use of heuristics which makes them slow and sometimes practically not reasonable. Another approach employs the greedy algorithm for obtaining available edge disjoint paths. Even though it is fast, it produces a solution for any connection request which is far from the optimal utilization of wavelengths. We propose a novel algorithm which is based on the maximum flow to have the maximum quantity of edge disjoint paths. Comprehensive computer simulation shows that the proposed method outperforms previous ones significantly in terms of running time. Furthermore, it shows compatible or better performance comparing to others in number of wavelengths used.

WDM방식을 기반으로 한 광 네트워크상에서 최대 EDPs(Edge Disjoint Paths)을 이용한 라우팅 및 파장할당 알고리즘

본 논문에서 파장 분할 다중화(WDM) 방식을 이용한 광 네트워크상의 라우팅과 파장할당 알고리즘을 고찰해 보겠다. 선형 프로그래밍(Linear Programming)과 그래프 컬러링(Graph Coloring)의 조합으로 이루어진 기존의 RWA기법들은 복잡하며, 발견적 방법(Heuristic Method) 사용이 요구된다. 이와 같은 방법은 실행시간이 길며, 최악의 경우에는 실행이 불가능하여 결과를 얻지 못한다. RWA를 해결하기 위한 다른 방법은 최대 EDPs(Edge Disjoint Paths)를 얻기 위해 greedy algorithm을 적용하는 것이다. 이것은 실행시간이 짧지만 파장의 수를 최적으로 사용하지 못한다. 본 논문에서 최대의 EDPs를 얻기 위해서 최대 흐름 기법(Maximum Flow Technique)을 이용한 새로운 알고리즘을 제안한다. 그리고 제안한 알고리즘과 기존에 제시된 최대 EDPs 알고리즘을 비교해 보겠다. In the present paper routing and wavelength assignment (RWA) in optical WDM networks is considered. Previous techniques based on the combination of integer linear programming and graph coloring are complex and require extensive use of heuristics. Such methods are mostly slow and sometimes impossible to get results due to infeasibility. An alternative approach applied to RWA employs on the greedy algorithm for obtaining the maximum edge disjoint paths. Even though this approach is fast, it produces a solution for any connection request, which is very far from the optimal utilization of wavelengths. We propose a novel algorithm, which is based on the maximum flow technique to obtain the maximum quantity of edge, disjoint paths. Here we compare the offered method with previous maximum edge disjoint paths algorithms ap plied to the RWA.

Linear coloring of graphs

A proper vertex coloring of a graph is called linear if the subgraph induced by the vertices colored by any two colors is a set of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by lc( G), is the minimum number of colors in a linear coloring of G. Extending a result of Alon, McDiarmid and Reed concerning acyclic graph colorings, we show that if G has maximum degree d then lc( G) = O( d 3/2). We also construct explicit graphs with maximum degree d for which lc(G) = Ω (d 3/2) , thus showing that the result is optimal, up to an absolute constant factor.