Linear Chromatic Number Research Articles

The linear chromatic number of a Sperner family

Let S be a finite set and S a complete Sperner family on S, i.e. a Sperner family such that every x∈S is contained in some member of S. The linear chromatic number of S, defined by Cıvan, is the smallest integer n with the property that there exists a function f:S→{1,…,n} such that if f(x)=f(y), then every set in S which contains x also contains y or every set in S which contains y also contains x. We give an explicit formula for the number of complete Sperner families on S of linear chromatic number 2. We also prove tight bounds on the number of elements in a Sperner family of given chromatic number, and prove that complete Sperner families of maximum linear chromatic number are far more numerous those of lesser linear chromatic number.

The Linear t-Colorings of Sierpiński-Like Graphs

A proper t-coloring of a graph G is a mapping $${\varphi: V(G) \rightarrow [1, t]}$$ such that $${\varphi(u) \neq \varphi(v)}$$ if u and v are adjacent vertices, where t is a positive integer. The chromatic number of a graph G, denoted by $${\chi(G)}$$ , is the minimum number of colors required in any proper coloring of G. A linear t-coloring of a graph is a proper t-coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by $${lc(G)}$$ , is the minimum t such that G has a linear t-coloring. In this paper, the linear t-colorings of Sierpinski-like graphs S(n, k), $${S^+(n, k)}$$ and $${S^{++}(n, k)}$$ are studied. It is obtained that $${lc(S(n, k))= \chi (S(n, k)) = k}$$ for any positive integers n and k, $${lc(S^+(n, k)) = \chi(S^+(n, k)) = k}$$ and $${lc(S^{++}(n, k)) = \chi(S^{++}(n, k)) = k}$$ for any positive integers $${n \geq 2}$$ and $${k \geq 3}$$ . Furthermore, we have determined the number of paths and the length of each path in the subgraph induced by the union of any two color classes completely.

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 ...

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.

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

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.

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.