Maximum Average Degree Research Articles

Tuza’s Conjecture for graphs with maximum average degree less than [formula omitted

Tuza’s Conjecture states that if a graph G does not contain more than k edge-disjoint triangles, then some set of at most 2k edges meets all triangles of G. We prove Tuza’s Conjecture for all graphs G having no subgraph with average degree at least 7. As a key tool in the proof, we introduce a notion of reducible sets for Tuza’s Conjecture; these are substructures which cannot occur in a minimal counterexample to Tuza’s Conjecture. We also introduce weak König–Egerváry graphs, a generalization of the well-studied König–Egerváry graphs.

On strong list edge coloring of subcubic graphs

In this paper, we give some new results on strong list edge coloring of subcubic graphs. We prove that every subcubic graph with maximum average degree less than 52, 83 and 145 can be strongly list edge colored with seven, eight and nine colors respectively.

Improper coloring of sparse graphs with a given girth, I: (0,1)-colorings of triangle-free graphs

A graph G is (0,1)-colorable if V(G) can be partitioned into two sets V0 and V1 so that G[V0] is an independent set and G[V1] has maximum degree at most 1. The problem of verifying whether a graph is (0,1)-colorable is NP-complete even in the class of planar graphs of girth 9.The maximum average degree, mad(G), of a graph G is the maximum of 2|E(H)||V(H)| over all subgraphs H of G. It was proved recently that every graph G with mad(G)≤125 is (0,1)-colorable, and this is sharp. This yields that every planar graph with girth at least 12 is (0,1)-colorable.Let F(g) denote the supremum of a such that for some constant bg every graph G with girth g and |E(H)|≤a|V(H)|+bg for every H⊆G is (0,1)-colorable. By the above, F(3)=1.2. We find the exact value for F(4) and F(5): F(4)=F(5)=119. In fact, we also find the best possible values of b4 and b5: every triangle-free graph G with |E(H)|<11|V(H)|+59 for every H⊆G is (0,1)-colorable, and there are infinitely many not (0,1)-colorable graphs G with girth 5, |E(G)|=11|V(G)|+59 and |E(H)|<11|V(H)|+59 for every proper subgraph H of G. A corollary of our result is that every planar graph of girth 11 is (0,1)-colorable. This answers a half of a question by Dorbec, Kaiser, Montassier and Raspaud. In a companion paper, we show that for every g, F(g)≤1.25 and resolve some similar problems for the so-called (j,k)-colorings generalizing (0,1)-colorings.

Técnicas radiográficas para pacientes pediátricos com necessidades especiais na Odontologia

Patients with special needs are usually dependent on family to aid oral hygiene, there is greater risk of exposure to dental caries. To carry out proper planning, it is necessary to obtain a correct diagnosis. The radiograph is one of the important methods for the completion of the diagnosis. However, so as to achieve a greater potential for information to a radiographic image, the same must be present technically perfect, ie, minimal distortion and maximum average degree of detail and density and contrast. The aim of this work is to demonstrate a way of running intraoral radiographs in patients with special needs with satisfactory level and medium of collaboration, promoting emotional support during the procedure.

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given (hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree 4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) ⩽ 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.

List coloring the square of sparse graphs with large degree

We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors.Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree Δ are list (Δ+1)-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Euler’s Formula, planar graphs of girth at least 6 are of maximum average degree less than 3, and planar graphs of girth at least 7 are of maximum average degree less than 145<3.We strengthen their result and prove that there exists a function f such that the square of any graph with maximum average degree m<3 and maximum degree Δ≥f(m) is list (Δ+1)-colorable. Note the planarity assumption is dropped. This bound of 3 is optimal in the sense that the above-mentioned planar graphs with girth 6 have maximum average degree less than 3 and arbitrarily large maximum degree, while their square cannot be (Δ+1)-colored. The same holds for list injective Δ-coloring.

Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree

A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h] = {1, 2, …, h}. Let w(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. For each edge u v ∈ E(G), if w(u) ≠ w(v), then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G. By tndiΣ (G), we denote the smallest value h in such a coloring of G. In this paper, we obtain that G is a graph with at least two vertices, if mad(G) < 3, then tndiΣ (G) ≤ k + 2 where k = max{Δ(G), 5}. It partially confirms the conjecture proposed by Pilśniak and Woźniak.

2‐Distance Coloring of Sparse Graphs

AbstractFor graphs of bounded maximum average degree, we consider the problem of 2‐distance coloring, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. We prove that graphs with maximum average degree less than and maximum degree Δ at least 4 are 2‐distance ‐colorable, which is optimal and improves previous results from Dolama and Sopena, and from Borodin et al. We also prove that graphs with maximum average degree less than (resp. , ) and maximum degree Δ at least 5 (resp. 6, 8) are list 2‐distance ‐colorable, which improves previous results from Borodin et al., and from Ivanova. We prove that any graph with maximum average degree m less than and with large enough maximum degree Δ (depending only on m) can be list 2‐distance ‐colored. There exist graphs with arbitrarily large maximum degree and maximum average degree less than 3 that cannot be 2‐distance ‐colored: the question of what happens between and 3 remains open. We prove also that any graph with maximum average degree can be list 2‐distance ‐colored (C depending only on m). It is optimal as there exist graphs with arbitrarily large maximum degree and maximum average degree less than 4 that cannot be 2‐distance colored with less than colors. Most of the above results can be transposed to injective list coloring with one color less.

The 1,2,3-Conjecture and 1,2-Conjecture for sparse graphs

The 1, 2, 3-Conjecture states that the edges of a graph without isolated edges can be labeled from {1, 2, 3} so that the sums of labels at adjacent vertices are distinct. The 1, 2-Conjecture states that if vertices also receive labels and the vertex label is added to the sum of its incident edges, then adjacent vertices can be distinguished using only {1, 2}. We show that various configurations cannot occur in minimal counterexamples to these conjectures. Discharging then confirms the conjectures for graphs with maximum average degree less than 8/3. The conjectures are already confirmed for larger families, but the structure theorems and reducibility results are of independent interest.

Acyclic edge coloring of graphs

An acyclic edge coloring of a graph G is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The acyclic chromatic indexχa′(G) of a graph G is the least number of colors needed in an acyclic edge coloring of G. Fiamčík (1978) conjectured that χa′(G)≤Δ(G)+2, where Δ(G) is the maximum degree of G. This conjecture is well known as the Acyclic Edge Coloring Conjecture (AECC). A graph G with maximum degree at most κ is κ-deletion-minimal if χa′(G)>κ and χa′(H)≤κ for every proper subgraph H of G. The purpose of this paper is to provide many structural lemmas on κ-deletion-minimal graphs. By using the structural lemmas, we firstly prove that AECC is true for the graphs with maximum average degree less than four (Theorem 4.3). We secondly prove that AECC is true for the planar graphs without triangles adjacent to cycles of length at most four, with an additional condition that every 5-cycle has at most three edges contained in triangles (Theorem 4.4), from which we can conclude some known results as corollaries. We thirdly prove that every planar graph G without intersecting triangles satisfies χa′(G)≤Δ(G)+3 (Theorem 4.6). Finally, we consider one extreme case and prove it: if G is a graph with Δ(G)≥3 and all the 3+-vertices are independent, then χa′(G)=Δ(G). We hope the structural lemmas will shed some light on the acyclic edge coloring problems.

List backbone colouring of graphs

Suppose that G is a graph and that H is a subgraph of G. Let L be a mapping that assigns to each vertex v of G a set L(v) of positive integers. We say that (G,H) is backboneL-colourable if there is a proper vertex colouring c of G such that c(v)∈L(v) for all v∈V, and |c(u)−c(v)|⩾2 for every edge uv in H. We say that (G,H) is backbone k-choosable if (G,H) is backbone L-colourable for any list assignment L with |L(v)|=k for all v∈V(G). The backbone choice number of (G,H), denoted by chBB(G,H), is the minimum k such that (G,H) is backbone k-choosable. The concept of a backbone choice number is a generalization of both the choice number and the L(2,1)-choice number. Precisely, if E(H)=0̸, then chBB(G,H)=ch(G), where ch(G) is the choice number of G; if G=H2, then chBB(G,H) is the same as the L(2,1)-choice number of H. In this article, we first show that, if |L(v)|=dG(v)+2dH(v), then (G,H) is L-colourable, unless E(H)=0̸ and each block of G is a complete graph or an odd cycle. This generalizes a result of Erdős, Rubin, and Taylor on degree-choosable graphs. Second, we prove that chBB(G,H)⩽max{⌊mad(G)⌋+1,⌊mad(G)+2mad(H)⌋}, where mad(G) is the maximum average degree of a graph G. Finally, we establish various upper bounds on chBB(G,H) in terms of ch(G). In particular, we prove that, for a k-choosable graph G, chBB(G,H)⩽3k if every component of H is unicyclic; chBB(G,H)⩽2k if H is a matching; and chBB(G,H)⩽2k+1 if H is a disjoint union of paths with length at most 2.

Graphs with maximum degree [formula omitted] and maximum average degree less than [formula omitted] are list [formula omitted]-distance [formula omitted]-colorable

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. It is already known that planar graphs of girth at least 6 and of maximum degree Δ are list 2-distance (Δ+2)-colorable when Δ≥24 (Borodin and Ivanova (2009)) and 2-distance (Δ+2)-colorable when Δ≥18 (Borodin and Ivanova (2009)). We prove here that Δ≥17 suffices in both cases. More generally, we show that graphs with maximum average degree less than 3 and Δ≥17 are list 2-distance (Δ+2)-colorable. The proof can be transposed to list injective (Δ+1)-coloring.

Defective 2-colorings of sparse graphs

A graph G is (j,k)-colorable if its vertices can be partitioned into subsets V1 and V2 such that every vertex in G[V1] has degree at most j and every vertex in G[V2] has degree at most k. We prove that if k⩾2j+2, then every graph with maximum average degree at most 2(2−k+2(j+2)(k+1)) is (j,k)-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close to 2(2−k+2(j+2)(k+1)) (from above) that are not (j,k)-colorable.In fact, we prove a stronger result by establishing the best possible sufficient condition for the (j,k)-colorability of a graph G in terms of the minimum, φj,k(G), of the difference φj,k(W,G)=(2−k+2(j+2)(k+1))|W|−|E(G[W])| over all subsets W of V(G). Namely, every graph G with φj,k(G)>−1k+1 is (j,k)-colorable. On the other hand, we construct infinitely many non-(j,k)-colorable graphs G with φj,k(G)=−1k+1.

Neighbor sum distinguishing edge colorings of graphs with bounded maximum average degree

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k], where [k]={1,2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that, for each edge uv∈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∑(G), we denote the smallest value k in such a coloring of G. The average degree of a graph G is ∑v∈V(G)d(v)|V(G)|; we denote it by ad(G). The maximum average degree mad(G) of G is the maximum of average degrees of its subgraphs. In this paper, we show that, if G is a graph without isolated edges and mad(G)≤52, then ndi∑(G)≤k, where k=max{Δ(G)+1,6}. This partially confirms the conjecture proposed by Flandrin et al.

Sufficient sparseness conditions for [formula omitted] to be [formula omitted]-choosable, when [formula omitted

We determine the list chromatic number of the square of a graph χℓ(G2) in terms of its maximum degree Δ when its maximum average degree, denoted mad(G), is sufficiently small. For Δ≥6, if mad(G)<2+4Δ−85Δ+2, then χℓ(G2)=Δ+1. In particular, if G is planar with girth g≥7+12Δ−2, then χℓ(G2)=Δ+1. Under the same conditions, χℓi(G)=Δ, where χℓi is the list injective chromatic number.

On [formula omitted]-improper [formula omitted]-coloring of sparse graphs

A graph G is (1,1)-colorable if its vertices can be partitioned into subsets V1 and V2 such that every vertex in G[Vi] has degree at most 1 for each i∈{1,2}. We prove that every graph with maximum average degree at most 145 is (1,1)-colorable. In particular, it follows that every planar graph with girth at least 7 is (1,1)-colorable. On the other hand, we construct graphs with maximum average degree arbitrarily close to 145 (from above) that are not (1,1)-colorable.In fact, we establish the best possible sufficient condition for the (1,1)-colorability of a graph G in terms of the minimum, ρG, of ρG(S)=7|S|−5|E(G[S])| over all subsets S of V(G). Namely, every graph G with ρG≥0 is (1,1)-colorable. On the other hand, we construct infinitely many non-(1,1)-colorable graphs G with ρG=−1. This solves a related conjecture of Kurek and Ruciński from 1994.

INJECTIVELY DELTA CHOOSABLE GRAPHS

Abstract. An injective coloring of a graph G is an assignment of colorsto the vertices of G so that any two vertices with a common neighborreceive distinct colors. A graph G is said to be injectively k-choosableif any list L(v) of size at least k for every vertex v allows an injectivecoloring φ(v) such that φ(v) ∈ L(v) for every v ∈ V (G). The least k forwhich G is injectively k-choosable is the injective choosability number ofG, denoted by χ li (G). In this paper, we obtain new sufficient conditionsto be χ li (G) = ∆(G). Maximum average degree, mad(G), is defined bymad(G) = max{2e(H)/n(H) : H is a subgraph of G}. We prove that ifmad(G) < 8k−33k , then χ li (G) = ∆(G) where k = ∆(G) and ∆(G) ≥ 6. Inaddition, when ∆(G) = 5 we prove that χ li (G) = ∆(G) if mad(G) < 177 ,and when ∆(G) = 4 we prove that χ li (G) = ∆(G) if mad(G) < 73 . Theseresults generalize some of previous results in [1, 4]. 1. IntroductionAll graphsconsidered in this paper aresimple, finite, and undirected. We useV (G),E(G) and ∆(G) to denote the vertex set, the edge set and the maximumdegree of G, respectively. Here we introduce some notation. A k-vertex is avertex of degree k, and a k

On strong edge-colouring of subcubic graphs

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of length 3 uses three different colours. In this paper we improve some previous results on the strong edge-colouring of subcubic graphs by showing that every subcubic graph with maximum average degree strictly less than 73 (resp. 52, 83, 207) can be strongly edge-coloured with six (resp. seven, eight, nine) colours. These upper bounds are optimal except the one of 83. Also, we prove that every subcubic planar graph without 4-cycles and 5-cycles can be strongly edge-coloured with nine colours.

Two smaller upper bounds of list injective chromatic number

An injective coloring of a graph $$G$$ is an assignment of colors to the vertices of $$G$$ so that any two vertices with a common neighbor receive distinct colors. Let $$\chi _{i}^{l}(G)$$ denote the list injective chromatic number of $$G$$ . We prove that (1) $$\chi _{i}^{l}(G)=\Delta $$ for a graph $$G$$ with the maximum average degree $$Mad(G)\le \frac{18}{7}$$ and maximum degree $$\Delta \ge 9$$ ; (2) $$\chi _{i}^{l}(G)\le \Delta +2$$ if $$G$$ is a plane graph with $$\Delta \ge 21$$ and without 3-, 4-, 8-cycles.

Vertex Partitions of Graphs into Cographs and Stars

A cograph is a graph that contains no path on four vertices as an induced subgraph. A cograph k-partition of a graph G = (V,E) is a vertex partition of G into k sets V1, …, Vk ⊂ V so that the graph induced by Vi is a cograph for 1 ≤ i ≤ k. Gimbel and Nešetřil [5] studied the complexity aspects of the cograph k-partitions and raised the following questions: Does there exist a triangle-free planar graph that is not cograph 2-partitionable? If the answer is yes, what is the complexity of the associated decision problem? In this article, we prove that such an example exists and that deciding whether a triangle-free planar graph admits a cograph 2-partition is NP-complete. We also show that every graph with maximum average degree at most ??? admits a cograph 2-partition such that each component is a star on at most three vertices.