Backbone Coloring Research Articles

On λ $\lambda $‐backbone coloring of cliques with tree backbones in linear time

AbstractA ‐backbone coloring of a graph with its subgraph (also called a backbone) is a function ensuring that is a proper coloring of and for each it holds that . In this paper we propose a way to color cliques with tree and forest backbones in linear time that the largest color does not exceed . This result improves on the previously existing approximation algorithms as it is ‐absolutely approximate, that is, with an additive error over the optimum. We also present an infinite family of trees with for which the coloring of cliques with backbones requires at least colors for close to . The construction draws on the theory of Fibonacci numbers, particularly on Zeckendorf representations.

Backbone coloring of graphs with galaxy backbones

A (proper) k -coloring of a graph G = ( V , E ) is a function c : V ( G ) → { 1 , … , k } such that c ( u ) ≠ c ( v ) , for every u v ∈ E ( G ) . Given a graph G and a spanning subgraph H of G , a (circular) q -backbone k -coloring of ( G , H ) is a k -coloring c of G such that q ≤ | c ( u ) − c ( v ) | ( q ≤ | c ( u ) − c ( v ) | ≤ k − q ), for every edge u v ∈ E ( H ) . The (circular) q -backbone chromatic number of ( G , H ) , denoted by BBC q ( G , H ) ( CBC q ( G , H ) ), is the minimum integer k for which there exists a (circular) q -backbone k -coloring of ( G , H ) . In this work, we (partially) answer three questions posed by Havet et al. (2014), namely, we prove that if G is a planar graph, H is a spanning subgraph of G and q is a positive integer, then: CBC q ( G , H ) ≤ 2 q + 2 when q ≥ 3 and H is a galaxy; CBC q ( G , H ) ≤ 2 q when q ≥ 4 and H is a matching; and CBC 3 ( G , H ) ≤ 7 when H is a matching and G has no triangles sharing an edge. In addition, we present a polynomial-time algorithm to determine both parameters for any pair ( G , H ) , whenever G has bounded treewidth. Finally, we show how to fix a mistake in a proof that BBC 2 ( G , M ) ≤ Δ ( G ) + 1 , for any matching M of an arbitrary graph G (Miškuf et al., 2010).

On the Existence of Tree Backbones that Realize the Chromatic Number on a Backbone Coloring

AbstractA proper k‐coloring of a graph is a function such that , for every . The chromatic number is the minimum k such that there exists a proper k‐coloring of G. Given a spanning subgraph H of G, a q‐backbone k‐coloring of is a proper k‐coloring c of such that , for every edge . The q‐backbone chromatic number is the smallest k for which there exists a q‐backbone k‐coloring of . In this work, we show that every connected graph G has a spanning tree T such that , and that this value is the best possible. As a direct consequence, we get that every connected graph G has a spanning tree T for which , if , or , otherwise. Thus, by applying the Four Color Theorem, we have that every connected nonbipartite planar graph G has a spanning tree T such that . This settles a question by Wang, Bu, Montassier, and Raspaud (J Combin Optim 23(1) (2012), 79–93), and generalizes a number of previous partial results to their question.

Optimal backbone coloring of split graphs with matching backbones

Optimal backbone coloring of split graphs with matching backbones

The Backbone Coloring Problem for Bipartite Backbones

Let $$G$$G be a simple graph, $$H$$H be its spanning subgraph and $$\lambda \ge 2$$??2 be an integer. By a $$\lambda $$?-backbone coloring of $$G$$G with backbone $$H$$H we mean any function $$c$$c that assigns positive integers to vertices of $$G$$G in such a way that $$|c(u)-c(v)|\ge 1$$|c(u)-c(v)|?1 for each edge $$uv\in E(G)$$uv?E(G) and $$|c(u)-c(v)|\ge \lambda $$|c(u)-c(v)|?? for each edge $$uv\in E(H)$$uv?E(H). The $$\lambda $$?-backbone chromatic number $$BBC_\lambda (G,H)$$BBC?(G,H) is the smallest integer $$k$$k such that there exists a $$\lambda $$?-backbone coloring $$c$$c of $$G$$G with backbone $$H$$H satisfying $$\max c(V(G))=k$$maxc(V(G))=k. A $$\lambda $$?-backbone coloring $$c$$c of $$G$$G with backbone $$H$$H is optimal if and only if $$\max c(V(G))=BBC_\lambda (G,H)$$maxc(V(G))=BBC?(G,H). In the paper we study the problem of finding optimal $$\lambda $$?-backbone colorings of complete graphs with bipartite backbones. We present a linear algorithm that is $$2$$2-approximate in general and $$1.5$$1.5-approximate if backbone is connected. Next we show a quadratic algorithm for backbones being trees that finds optimal $$\lambda $$?-backbone colorings provided $$\lambda $$? is large enough.

Backbone coloring of planar graphs without special circles

In this paper, we prove that if G is a connected planar graph that is C6-free or C7-free and without adjacent triangles, then there exists a spanning tree T of G such that χb(G,T)≤4.

On backbone coloring of graphs

Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G,H) is a mapping f: V(G)?{1,2,?,k} such that |f(u)?f(v)|?2 if uv?E(H) and |f(u)?f(v)|?1 if uv?E(G)\E(H). The backbone chromatic number of (G,H) is the smallest integer k such that (G,H) has a backbone-k-coloring. In this paper, we characterize the backbone chromatic number of Halin graphs G=T?C with respect to given spanning trees T. Also we study the backbone coloring for other special graphs such as complete graphs, wheels, graphs with small maximum average degree, graphs with maximum degree 3, etc.

Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G ), a backbone coloring for G and H is a proper vertex k -coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2 . The minimal k ∈ N for which such a coloring exists is called the backbone chromatic number of G . We show that for a graph G of maximum degree Δ where the backbone graph is a d -degenerated subgraph of G , the backbone chromatic number is at most Δ + d + 1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most Δ + 1 . We also present examples where these bounds are attained. Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H . We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G , then the backbone chromatic number is at most Δ ( G ) − Δ ( G ) .

Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number

We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph $G = (V,E)$ and a spanning subgraph $H$ of $G$ (the backbone of $G$), a $\gamma$-backbone coloring for $G$ and $H$ is a proper vertex coloring $V\to \{1,2,\ldots\}$ of $G$ in which the colors assigned to adjacent vertices in $H$ differ by at least $\gamma$. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers. The main outcome of earlier studies is that the minimum number $\ell$ of colors, for which such colorings $V\to \{1,2,\ldots, \ell\}$ exist, in the worst case is a factor times the chromatic number (for path, tree, matching and star backbones). We show here that for split graphs and matching or star backbones, $\ell$ is at most a small additive constant (depending on $\gamma$) higher than the chromatic number. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on $\ell$ than the previously known bounds.

Formula omitted]-backbone colorings along pairwise disjoint stars and matchings

Given an integer λ ≥ 2 , a graph G = ( V , E ) and a spanning subgraph H of G (the backbone of G ), a λ -backbone coloring of ( G , H ) is a proper vertex coloring V → { 1 , 2 , … } of G , in which the colors assigned to adjacent vertices in H differ by at least λ . We study the case where the backbone is either a collection of pairwise disjoint stars or a matching. We show that for a star backbone S of G the minimum number ℓ for which a λ -backbone coloring of ( G , S ) with colors in { 1 , … , ℓ } exists can roughly differ by a multiplicative factor of at most 2 − 1 λ from the chromatic number χ ( G ) . For the special case of matching backbones this factor is roughly 2 − 2 λ + 1 . We also show that the computational complexity of the problem “Given a graph G with a star backbone S , and an integer ℓ , is there a λ -backbone coloring of ( G , S ) with colors in { 1 , … , ℓ } ?” jumps from polynomially solvable to NP-complete between ℓ = λ + 1 and ℓ = λ + 2 (the case ℓ = λ + 2 is even NP-complete for matchings). We finish the paper by discussing some open problems regarding planar graphs.

Backbone colorings for graphs: Tree and path backbones

AbstractWe introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1,2,…} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number χ(G); for path backbones this factor is roughly ${{3}\over{2}}$. We show that the computational complexity of the problem “Given a graph G, a spanning tree T of G, and an integer ℓ, is there a backbone coloring for G and T with at most ℓ colors?” jumps from polynomial to NP‐complete between ℓ = 4 (easy for all spanning trees) and ℓ = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 137–152, 2007