D-degenerate Graphs Research Articles

Ramsey numbers of degenerate graphs

A graph is d-degenerate if all its subgraphs have a vertex of degree at most d. We prove that there exists a constant c such that for all natural numbers d and r, every d-degenerate graph G of chromatic number r has Ramsey number at most 2 d2 cr |V (G)|. This solves a conjecture of Burr and Erdýos from 1973.

On complexities of minus domination

A function f:V→{−1,0,1} is a minus-domination function of a graph G=(V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x∈V. In the minus-domination problem, one tries to minimize the weight of a minus-domination function. In this paper, we show that (1) the minus-domination problem is fixed-parameter tractable for d-degenerate graphs when parameterized by the size of the minus-dominating set and by d, where the size of a minus domination is the number of vertices that are assigned 1, (2) the minus-domination problem is polynomial for graphs of bounded rankwidth and for strongly chordal graphs, (3) it is NP-complete for split graphs, and (4) there is no fixed-parameter algorithm for minus-domination.

K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

A K-3-WORM coloring of a graph G is an assignment of colors to the vertices in such a way that the vertices of each K-3-subgraph of G get precisely two colors. We study graphs G which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer. 219 (2014) 161-173] by proving that for every integer k >= 3 there exists a K-3-WORM-colorable graph in which the minimum number of colors is exactly k. There also exist K-3-WORM colorable graphs which have a K-3-WORM coloring with two colors and also with k colors but no coloring with any of 3, ..., k - 1 colors. We also prove that it is NP-hard to determine the minimum number of colors, and NP-complete to decide k-colorability for every k >= 2 (and remains intractable even for graphs of maximum degree 9 if k = 3). On the other hand, we prove positive results for d-degenerate graphs with small d, also including planar graphs.

Neighbor Sum (Set) Distinguishing Total Choosability of d-Degenerate Graphs

Let $$G=(V,E)$$G=(V,E) be a graph with maximum degree $$\varDelta (G)$$Δ(G) and $$\phi :V\cup E\rightarrow \{1,2,\ldots ,k\}$$?:V?E?{1,2,?,k} be a proper total coloring of the graph G. Let S(v) denote the set of the color on vertex v and the colors on the edges incident with v. Let f(v) denote the sum of the color on vertex v and the colors on the edges incident with v. The proper total coloring $$\phi $$? is called neighbor set distinguishing or adjacent vertex distinguishing if $$S(u)\ne S(v)$$S(u)?S(v) for each edge $$uv\in E(G)$$uv?E(G). We say that $$\phi $$? is neighbor sum distinguishing if $$f(u)\ne f(v)$$f(u)?f(v) for each edge $$uv\in E(G)$$uv?E(G). In both problems the challenging conjectures presume that such colorings exist for any graph G if $$k\ge \varDelta (G)+3$$k?Δ(G)+3. Ding et al. proved in both problems $$k\ge \varDelta (G)+2d$$k?Δ(G)+2d is sufficient for d-degenerate graph G. In this paper, we improve this bound and prove that $$k\ge \varDelta (G)+d+1$$k?Δ(G)+d+1 is sufficient for d-degenerate graph G with $$d\le 8$$d≤8 and $$\varDelta (G)\ge 2d$$Δ(G)?2d or $$d\ge 9$$d?9 and $$\varDelta (G)\ge \frac{5}{2}d-5$$Δ(G)?52d-5. In fact, we prove these results in their list versions. As a consequence, we obtain an upper bound of the form $$\varDelta (G)+C$$Δ(G)+C for some families of graphs, e.g. $$\varDelta (G)+6$$Δ(G)+6 for planar graphs with $$\varDelta (G)\ge 10$$Δ(G)?10. In particular, we therefore obtain that when $$\varDelta (G)\ge 4$$Δ(G)?4 two conjectures we mentioned above hold for 2-degenerate graphs in their list versions.

Fast recoloring of sparse graphs

In this paper, we show that for every graph of maximum average degree bounded away from d and any two (d+1)-colorings of it, one can transform one coloring into the other one within a polynomial number of vertex recolorings so that, at each step, the current coloring is proper. In particular, it implies that we can transform any 8-coloring of a planar graph into any other 8-coloring with a polynomial number of recolorings. These results give some evidence on a conjecture of Cereceda et al. (2009) which asserts that any (d+2) coloring of a d-degenerate graph can be transformed into any other one using a polynomial number of recolorings.We also show that any (2d+2)-coloring of a d-degenerate graph can be transformed into any other one with a linear number of recolorings.

Equitable partition of graphs into induced forests

An equitable partition of a graph G is a partition of the vertex-set of G such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most k colors can be equitably partitioned into k−1 induced forests. We also prove that for any integers d⩾1 and k⩾3d−1, any d-degenerate graph can be equitably partitioned into k induced forests.Each of these results implies the existence of a constant c such that for any k⩾c, any planar graph has an equitable partition into k induced forests. This was conjectured by Wu, Zhang, and Li in 2013.

On almost [formula omitted]-degenerate [formula omitted]-chromatic graphs and hypergraphs

Recall that a (hyper)graph is d-degenerate if each of its nonempty subgraphs has a vertex of degree at most d. Every d-degenerate (hyper)graph is (easily) (d+1)-colorable. A (hyper)graph is almostd-degenerate if it is not d-degenerate, but each of its proper subgraphs is d-degenerate. In particular, if G is almost (k−1)-degenerate, then after deleting any edge it is k-colorable. For k≥2, we study properties of almost (k−1)-degenerate (hyper)graphs that are not k-colorable. By definition, each such (hyper)graph is (k+1)-critical.

Kernelization hardness of connectivity problems in [formula omitted]-degenerate graphs

A graph is d-degenerate if every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by the recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for Dominating Set in d-degenerate graphs, we investigate the kernelization complexity of problems that include a connectivity requirement in this class of graphs.Our main contribution is the proof that Connected Dominating Set does not admit a polynomial kernel in d-degenerate graphs for d≥2 unless NP⊆coNP/poly, which is known to cause a collapse of the polynomial hierarchy to its third level. We prove this using a problem that originates from bioinformatics–Colourful Graph Motif–analysed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique also yields an alternative proof that, under the same complexity assumption, Steiner Tree does not admit a polynomial kernel. The original proof, via a reduction from Set Cover, is due to Dom, Lokshtanov and Saurabh.We extend our analysis by showing that, unless NP⊆coNP/poly, there do not exist polynomial kernels for Steiner Tree, Connected Feedback Vertex Set and Connected Odd Cycle Transversal in d-degenerate graphs for d≥2. On the other hand, we show a polynomial kernel for Connected Vertex Cover in graphs that do not contain the biclique Ki,j as a subgraph. As a d-degenerate graph cannot contain Kd+1,d+1 as a subgraph, the results holds also for graphs of bounded degeneracy.

Spectrally degenerate graphs: Hereditary case

It is well known that the spectral radius of a tree whose maximum degree is Δ cannot exceed 2Δ−1. A similar upper bound holds for arbitrary planar graphs, whose spectral radius cannot exceed 8Δ+10, and more generally, for all d-degenerate graphs, where the corresponding upper bound is 4dΔ. Following this, we say that a graph G is spectrally d-degenerate if every subgraph H of G has spectral radius at most dΔ(H). In this paper we derive a rough converse of the above-mentioned results by proving that each spectrally d-degenerate graph G contains a vertex whose degree is at most 4dlog2(Δ(G)/d) (if Δ(G)⩾2d). It is shown that the dependence on Δ in this upper bound cannot be eliminated, as long as the dependence on d is subexponential. It is also proved that the problem of deciding if a graph is spectrally d-degenerate is co-NP-complete.

First-fit chromatic numbers of [formula omitted]-degenerate graphs

The first-fit chromatic number of a graph is the number of colors needed in the worst case of a greedy coloring. In this short note, we first give counterexamples to some results by Balogh et al. (2008) [1], and then prove that every n-vertex d-degenerate graph has first-fit chromatic number at most logd+1dn+2.

Primality of trees

A graph of order n is prime if one can bijectively label its vertices with integers 1, . . . , n so that any two adjacent vertices get coprime labels. We prove that all bipartite d-degenerate graphs with separators of size at most n1−Od(1/ ln lnn) are prime. It immediately follows that all large trees are prime, confirming an old conjecture of Entringer and Tout from around 1980. Also, our method allows us to determine the smallest size of a non-prime connected order-n graph for all large n, proving a conjecture of Rao [R. C. Bose Centenary Symposium on Discrete Math. and Applications, Kolkata, 2002] in this range.

Packing d-degenerate graphs

We study packings of graphs with given maximal degree. We shall prove that the (hitherto unproved) Bollobás–Eldridge–Catlin Conjecture holds in a considerably stronger form if one of the graphs is d-degenerate for d not too large: if d , Δ 1 , Δ 2 ⩾ 1 and n > max { 40 Δ 1 ln Δ 2 , 40 d Δ 2 } then a d-degenerate graph of maximal degree Δ 1 and a graph of order n and maximal degree Δ 2 pack. We use this result to show that, for d fixed and n large enough, one can pack n 1500 d 2 arbitrary d-degenerate n-vertex graphs of maximal degree at most n 1000 d ln n .

On Equitable Coloring of d-Degenerate Graphs

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most 1. A d-degenerate graph is a graph G in which every subgraph has a vertex with degree at most d. A star Sm with m rays is an example of a 1-degenerate graph with maximum degree m that needs at least 1+m/2 colors for an equitable coloring. Our main result is that every n-vertex d-degenerate graph G with maximum degree at most n/15 can be equitably k-colored for each $k \ge 16d$. The proof of this bound is constructive. We extend the algorithm implied in the proof to an O(d)-factor approximation algorithm for equitable coloring of an arbitraryd -degenerate graph. Among the implications of this result is an O(1)-factor approximation algorithm for equitable coloring of planar graphs with fewest colors. A variation of equitable coloring (equitable partitions) is also discussed.

On Graphs With Small Ramsey Numbers, II

There exists a constant C such that for every d-degenerate graphs G1 and G2 on n vertices, Ramsey number R(G1,G2) is at most CnΔ, where Δ is the minimum of the maximum degrees of G1 and G2.

On Ramsey Numbers of Sparse Graphs

The Ramsey number, , of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdős posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(d) such that for any d-degenerate graph G of order n.In this paper we prove that for each such G. In fact, we show that, for every , sufficiently large n, and any graph H of order , either H or its complement contains a (d,n)-common graph, that is, a graph in which every set of d vertices has at least n common neighbours. It is easy to see that any (d,n)-common graph contains every d-degenerate graph G of order n. We further show that, for every constant C, there is an n and a graph H of order such that neither H nor its complement contains a -common graph.

Equitable Colourings of d-degenerate Graphs

A proper vertex coloring of a graph is called equitable if the sizes of colour classes differ by at most 1. In this paper, we find the minimum number l=l(d, Δ) such that every d-degenerate graph with maximum degree at most Δ admits an equitable t-colouring for every t[ges ]l when Δ[ges ]27d.