Maximum Average Degree Research Articles

Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers

Let G be a graph and ϕ:V(G)∪E(G)→{1,2,3,…,k} be a k-total coloring. Let w(v) denote the sum of color on a vertex v and colors assigned to edges incident to v. If w(u)≠w(v) whenever uv∈E(G), then ϕ is called a neighbor sum distinguishing total coloring. The smallest integer k such that G has a neighbor sum distinguishing k-total coloring is denoted by tndi∑ (G). In 2014, Dong and Wang obtained the results about tndi∑ (G) depending on the value of maximum average degree. A k-assignment L of G is a list assignment L of integers to vertices and edges with L(v)=k for each vertex v and L(e)=k for each edge e. A total-L-coloring is a total coloring ϕ of G such that ϕ(v)∈L(v) whenever v∈V(G) and ϕ(e)∈L(e) whenever e∈E(G). We state that G has a neighbor sum distinguishing total-L-coloring if G has a total-L-coloring such that w(u)≠w(v) for all uv∈E(G). The smallest integer k such that G has a neighbor sum distinguishing total-L-coloring for every k-assignment L is denoted by Ch∑ ′′(G). In this paper, we strengthen results by Dong and Wang by giving analogous results for Ch∑ ′′(G).

PS—Silica Hybrid Particle Prepared by Mechanochemical Method and the Study of Its Performance

ABSTRACTIn this paper, the polystyrene–silica hybrid particles were prepared by mechanochemical method. The surface activity index of the hybrid particle was prepared by mechanochemical method, and the contact angle is 128.8°, which indicates that it has the very good hydrophobicity. We can observe that the surface of the modified SiO2 coated with a layer of polystyrene under the transmission electron microscopy. In infrared, we found that there are Si–C bonds on the surface of the modified SiO2, which indicates that styrene monomer is successfully grafted to the surface of silica in the form of polystyrene whose maximum viscosity average degree of polymerization is 158 under mechanical force. The grafting rate was 4.28% measured by thermogravimetric analysis.

Total weight choosability of graphs with bounded maximum average degree

A total weighting of a graph G is a function ϕ that assigns a weight to each vertex and each edge of G. The vertex-sum of a vertex v with respect to ϕ is Sϕ(v)=ϕ(v)+∑e∈E(v)ϕ(e), where E(v) is the set of edges incident to v. A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph G is (k,k′)-choosable if the following is true: Whenever each vertex x is assigned a set L(x) of k real numbers and each edge e is assigned a set L(e) of k′ real numbers, there is a proper total weighting ϕ of G with ϕ(y)∈L(y) for all y∈V(G)∪E(G). In this paper, we prove that for p∈{5,7,11}, a graph G without isolated edges and with mad(G)≤p−1 is (1,p)-choosable. In particular, triangle-free planar graphs are (1,5)-choosable.

On weight choosabilities of graphs with bounded maximum average degree

The well known 1–2–3-Conjecture asserts that every connected graph G of order at least three can be edge-coloured with integers 1,2,3 so that the sums of colours met by adjacent vertices are distinct in G. The same is believed to hold if instead of edge colourings we consider total colourings assigning 1 or 2 to every vertex and edge of a given graph—this time the colour of every vertex is counted in its corresponding sum. We consider list extensions of the both concepts, where every edge (and vertex) is assigned a set of k positive integers, i.e., its potential colours, and regardless of this list assignment we wish to be able to construct a colouring from these lists so that the adjacent vertices are distinguished by their corresponding sums. We prove that if the maximum average degree of the graph G is smaller than 52, then lists of length k=3 are sufficient for that goal in case of edge colourings (if G has no isolated edges), while already k=2 suffices in the total case. In fact the second of these statements remains true with arbitrary real colours admitted instead of positive integers, and the first one—for positive reals. The proofs of these facts are based on the discharging method combined with the algebraic approach of Alon known as Combinatorial Nullstellensatz.

Star Chromatic Index of Subcubic Graphs

The star chromatic index of a graph G is the minimum number of colors needed to properly color the edges of G so that no path or cycle of length four is bi-colored. In this paper, we show that every subcubic graph with maximum average degree less than 115 can be star edge colored with at most five colors.

Coloring, sparseness and girth

An r-augmented tree is a rooted tree plus r edges added from each leaf to ancestors. For d, g, r ∈ N, we construct a bipartite r-augmented complete d-ary tree having girth at least g. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-k-choosable bipartite graphs, showing that maximum average degree at most 2(k - 1) is a sharp sufficient condition for k-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-k-colorable graphs and hypergraphs with any girth.

Adjacent vertex distinguishing total colorings of 2-degenerate graphs

Let ϕ be a proper total coloring of G. We use Cϕ(v)={ϕ(v)}∪{ϕ(uv)∣uv∈E(G)} to denote the set of colors assigned to a vertex v and those edges incident with v. An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that Cϕ(u)≠Cϕ(v) for any uv∈E(G). The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by χa″(G). In this paper we show that if G is a 2-degenerate graph, then χa″(G)≤max{Δ(G)+2,6}. Moreover, we also show that when Δ≥5, χa″(G)=Δ(G)+2 if and only if G contains two adjacent vertices of maximum degree. Our results imply the results on outerplanar graphs (Wang and Wang, 2010), K4-minor free graphs (Wang and Wang, 2009) and graphs with maximum average degree less than 3 (Wang and Wang, 2008).

Neighbor distinguishing total choice number of sparse graphs via the Combinatorial Nullstellensatz

Let G = (V,E) be a graph and ϕ: V ∪E → {1, 2, · · ·, k} be a total-k-coloring of G. Let f(v)(S(v)) denote the sum(set) of the color of vertex v and the colors of the edges incident with v. The total coloring ϕ is called neighbor sum distinguishing if (f(u) ≠ f(v)) for each edge uv ∈ E(G). We say that ϕ is neighbor set distinguishing or adjacent vertex distinguishing if S(u) ≠ S(v) for each edge uv ∈ E(G). For both problems, we have conjectures that such colorings exist for any graph G if k ≥ Δ(G) + 3. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs, which is denoted by mad (G). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that these two conjectures hold for sparse graphs in their list versions. More precisely, we prove that every graph G with maximum degree Δ(G) and maximum average degree mad(G) has chΣ″(G) ≤ Δ(G) + 3 (where chΣ″(G) is the neighbor sum distinguishing total choice number of G) if there exists a pair \((k,m) \in \{ (6,4),(5,\tfrac{{18}} {5}),(4,\tfrac{{16}} {5})\}\) such that Δ(G) ≥ k and mad (G) <m.

I,F-partitions of sparse graphs

A stark-coloring is a proper k-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than 52 has an I,F-partition, which is sharp and answers a question of Cranston and West (0000). This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu et al. (2009).

$$P_{11}$$ P 11 -Coloring of Oriented Graphs

We show that, every oriented graph with maximum average degree less than 28/9 admits a homomorphism into the Paley tournament with 11 vertices.

Coloring the square of graphs whose maximum average degree is less than 4

The square G2 of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. The maximum average degree of G, mad(G), is the maximum among the average degrees of the subgraphs of G.It is known in Bonamy et al. (2014) that there is no constant C such that every graph G with mad(G)<4 has χ(G2)≤Δ(G)+C. Charpentier (2014) conjectured that there exists an integer D such that every graph G with Δ(G)≥D and mad(G)<4 has χ(G2)≤2Δ(G). Recent result in Bonamy et al. (2014) [2] implies that χ(G2)≤2Δ(G) if mad(G)<4−1c with Δ(G)≥40c−16.In this paper, we show for an integer c≥2, if mad(G)<4−1c and Δ(G)≥14c−7, then χℓ(G2)≤2Δ(G), which improves the result in Bonamy et al. (2014) [2]. We also show that for every integer D, there is a graph G with Δ(G)≥D such that mad(G)<4, and χ(G2)=2Δ(G)+2, which disproves Charpentier’s conjecture. In addition, we give counterexamples to Charpentier’s another conjecture in Charpentier (2014), stating that for every integer k≥3, there is an integer Dk such that every graph G with mad(G)<2k and Δ(G)≥Dk has χ(G2)≤kΔ(G)−k.

On Equitable Colorings of Sparse Graphs

A graph is equitably k-colorable if G has a proper vertex k-coloring such that the sizes of any two color classes differ by at most one. Chen, Lih and Wu conjectured that any connected graph G with maximum degree $$\Delta $$ distinct from the odd cycle, the complete graph $$K_{\Delta +1}$$ and the complete bipartite graph $$K_{\Delta ,\Delta }$$ are equitably m-colorable for every $$m\ge \Delta $$ . Let $${\mathcal {G}}_k$$ be the class of graphs G such that $$e(G')\le k (v(G')-2)$$ for every subgraph $$G'$$ of G with order at least 3. In this paper, it is proved that any graph in $${\mathcal {G}}_4$$ with maximum degree $$\Delta \ge 17$$ is equitably m-colorable for every $$m\ge \Delta $$ . As corollaries, we confirm Chen–Lih–Wu Conjecture for 1-planar graphs, 3-degenerate graphs and graphs with maximum average degree less than 6, provided that $$\Delta \ge 17$$ .

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.

Adjacent vertex distinguishing colorings by sum of sparse graphs

A neighbor sum distinguishing edge-k-coloring, or nsd-k-coloring for short, of a graph G is a proper edge coloring of G with elements from {1,2,…,k} such that no pair of adjacent vertices meets the same sum of colors of G. The definition of this coloring makes sense for graphs containing no isolated edges (we call such graphs normal). Let mad(G) and Δ(G) be the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every normal graph with Δ(G)≥5 and mad(G)<3 admits an nsd-(Δ(G)+2)-coloring. Our approach is based on the Combinatorial Nullstellensatz and the discharging method.

Neighbor Sum Distinguishing Edge Colorings of Graphs with Small Maximum Average Degree

A proper edge-k-coloring of a graph G is an assignment of k colors $$1,2,\ldots ,k$$ to the edges of G such that no two adjacent edges receive the same color. A neighbor sum distinguishing edge-k-coloring of G is a proper edge-k-coloring of G such that for each edge $$uv\in E(G)$$ , the sum of colors taken on the edges incident with u is different from the sum of colors taken on the edges incident with v. By $${ {ndi}}_{\sum }(G)$$ , we denote the smallest value k in such a coloring of G. The maximum average degree of G is $${ {mad}}(G)=\max \{2|E(H)|/|V(H)|\}$$ , where the maximum is taken over all the non-empty subgraphs H of G. In this paper, we obtain that if G is a graph without isolated edges and $${ {mad}}(G)<8/3$$ , then $${ {ndi}}_{\sum }(G)\le k$$ where $$k=\max \{\Delta (G)+1,6\}$$ . It partially confirms the conjecture proposed by Flandrin et al. (Graphs Comb 29:1329–1336, 2013).

An Oriented 6-Coloring of Planar Graphs with Girth at Least 9

We prove that every oriented graph with a maximum average degree less than 18/7 admits a homomorphism into $$P_{7}^{*}$$P7?, the Paley tournament of order seven with one vertex deleted. In particular, every oriented planar graph of girth at least 9 has a homomorphism into $$P_{7}^{*}$$P7?, whence every planar graph of girth at least 9 has oriented chromatic number at most 6.

Universality of random graphs and rainbow embedding

AbstractIn this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of . Let the maximum density of a graph H be the maximum average degree of all the subgraphs of H. First, we show that for , a graph w.h.p. contains copies of all spanning graphs H with maximum degree at most Δ and maximum density at most d. For , this improves a result of Dellamonica, Kohayakawa, Rödl and Rucińcki. Next, we show that if we additionally restrict the spanning graphs to have girth at least 7 then the random graph contains w.h.p. all such graphs for . In particular, if , the random graph therefore contains w.h.p. every spanning tree with maximum degree bounded by Δ. This improves a result of Johannsen, Krivelevich and Samotij.Finally, in the same spirit, we show that for any spanning graph H with constant maximum degree, and for suitable p, if we randomly color the edges of a graph with colors, then w.h.p. there exists a rainbow copy of H in G (that is, a copy of H with all edges colored with distinct colors). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 546–564, 2016

The list 2-distance coloring of a graph with Δ(G) = 5

We study the list 2-distance coloring of a graph G, and its maximum average degree, denoted mad (G). For Δ(G) = 5, we proved that [Formula: see text], [Formula: see text], [Formula: see text] and ch 2(G) ≤ 11 if mad (G) &lt; 3 respectively, where ch 2(G) is the list 2-distance chromatic number.

Neighbor sum distinguishing edge colorings of sparse graphs

We consider proper edge colorings of a graph G using colors of the set {1,…,k}. Such a coloring is called neighbor sum distinguishing if for any uv∈E(G), the sum of colors of the edges incident to u is different from the sum of the colors of the edges incident to v. The smallest value of k in such a coloring of G is denoted by ndiΣ(G). Let mad(G) and Δ(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this paper we show that, for a graph G without isolated edges, if mad(G)<83, then ndiΣ(G)≤max{Δ(G)+1,7}; and if mad(G)<3, then ndiΣ(G)≤max{Δ(G)+2,7}.

Decomposing a graph into pseudoforests with one having bounded degree

The maximum average degree of a graph G, denoted by mad(G), is defined as mad(G)=maxH⊆G⁡2e(H)v(H). Suppose that σ is an orientation of G, Gσ denotes the oriented graph. It is well-known that for any graph G, there exists an orientation σ such that Δ+(Gσ)≤k if and only if mad(G)≤2k.A graph is called a pseudoforest if it contains at most one cycle in each component, is d-bounded if it has maximum degree at most d. In this paper, it is proven that, for any non-negative integers k and d, if G is a graph with mad(G)≤2k+2dk+d+1, then G decomposes into k+1 pseudoforests with one being d-bounded. This result in some sense is analogous to the Nine Dragon Tree (NDT) Conjecture, which is a refinement of the famous Nash–Williams Theorem that characterizes the decomposition of a graph into forests. A class of examples is also presented to show the sharpness of our result.