Maximum Average Degree Research Articles

Cliques in squares of graphs with maximum average degree less than 4

AbstractHocquard, Kim, and Pierron constructed, for every even integer , a 2‐degenerate graph with maximum degree such that . We prove for (a) all 2‐degenerate graphs and (b) all graphs with , upper bounds on the clique number of that match the lower bound given by this construction, up to small additive constants. We show that if is 2‐degenerate with maximum degree , then (with when is sufficiently large). And if has and maximum degree , then . Thus, the construction of Hocquard et al. is essentially the best possible. Our proofs introduce a “token passing” technique to derive crucial information about nonadjacencies in of vertices that are adjacent in . This is a powerful technique for working with such graphs that has not previously appeared in the literature.

The Spectral Radius, Maximum Average Degree and Cycles of Consecutive Lengths of Graphs

The Spectral Radius, Maximum Average Degree and Cycles of Consecutive Lengths of Graphs

Optimal Operation Model of Ecological Flow of Hydropower Station Based on Genetic Algorithm and Neural Network

In view of the contradiction between the ecological flow regulation of the reservoir River and the power generation flow regulation of the reservoir with the largest power generation, this paper takes the maximum power generation and the maximum average ecological protection degree as the objective function, and uses the genetic algorithm back propagation neural networks (GA-BP) algorithm to establish the reservoir flow regulation model under the constraints of water balance, reservoir capacity, unit output and unit overflow, The optimal solution of the dispatching scheme is carried out, and the Shulehe hydropower station is taken as an example to verify the optimal scheme. The results show that the optimized dispatching model can ensure the demand of ecological flow, increase 32680 kW·h power generation and 23.85% power generation benefit, so as to provide scientific reference for the rational planning, dispatching and utilization of reservoir water resources.

Equitable cluster partition of graphs with small maximum average degree

Equitable cluster partition of graphs with small maximum average degree

Neighbor Distinguishing Colorings of Graphs with the Restriction for Maximum Average Degree

Neighbor distinguishing colorings of graphs represent powerful tools for solving the channel assignment problem in wireless communication networks. They consist of two forms of coloring: neighbor distinguishing edge coloring, and neighbor distinguishing total coloring. The neighbor distinguishing edge (total) coloring of a graph G is an edge (total) coloring with the requirement that each pair of adjacent vertices contains different color sets. The neighbor distinguishing edge (total) chromatic number of G is the smallest integer k in cases where a neighbor distinguishing edge (total) coloring exists through the use of k colors in G. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs. In this paper, we characterize the neighbor distinguishing edge (total) chromatic numbers of graphs with a maximum average degree less than four by means of the discharging method.

On odd colorings of sparse graphs

An oddc-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing an odd number of times within its open neighborhood. A proper conflict-freec-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing exactly once within its neighborhood. Clearly, every proper conflict-free c-coloring is also an odd c-coloring. Cranston conjectured that every graph G with maximum average degree mad(G)<4cc+2 (where c≥4) has an odd c-coloring, and he proved this conjecture for c∈{5,6}. Note that the bound 4cc+2 is best possible. Cho et al. solved Cranston’s conjecture for c≥5, strengthening the result by transitioning from odd c-coloring to proper conflict-free c-coloring. However, they did not provide all the extremal non-colorable graphs G with mad(G)=4cc+2, which remains an open question of interest.In this paper, we tackle this intriguing extremal problem. We aim to characterize all non-proper conflict-free c-colorable graphs G with mad(G)=4cc+2. For the case of c=4, Cranston’s conjecture is not true, as evidenced by the existence of a counterexample: a graph whose every block is a 5-cycle. Cho et al. proved that a graph G with mad(G)<229 and no induced 5-cycles has an odd 4-coloring. We improve this result by proving that a graph G with mad(G)≤229 (with equality allowed) is not odd 4-colorable if and only if G belongs to a specific class of graphs. On the other hand, Cho et al. established that a planar graph with girth at least 5 has an odd 6-coloring; we improve it by proving that a planar graph without 4−-cycles adjacent to 7−-cycles also has an odd 6-coloring.

Towards the Chen-Raspaud conjecture

For an integer k, a homomorphism from a graph G to the Kneser graph K(2k+1,k) is equivalent to assigning to each vertex of G a k-subset of {1,…,2k+1} in a way that adjacent vertices receive disjoint subsets.Chen and Raspaud (2010) [5] conjectured that for every k≥2, every graph G with maximum average degree less than 2k+1k and no odd cycles with fewer than 2k+1 vertices admits a homomorphism to K(2k+1,k). They also showed that the statement is true for k=2. In this note we confirm the conjecture for k=3.

Kempe changes in degenerate graphs

We consider Kempe changes on the k-colorings of a graph on n vertices. If the graph is (k−1)-degenerate, then all its k-colorings are equivalent up to Kempe changes. However, the sequence between two k-colorings that arises from the proof may have length exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most k−1. Namely, any two k-colorings are equivalent up to O(kn2) Kempe changes. We investigate other restrictions (list coloring, bounded maximum average degree, degree bounds). As one of the main results, we derive that given an n-vertex graph with maximum degree Δ, the Δ-colorings are all equivalent up to OΔ(n2) Kempe changes, unless Δ=3 and some connected component is a 3-prism, that is K2□K3, in which case there exist some non-equivalent 3-colorings.

Digraph redicolouring

In this work, we generalize several results on graph recolouring to digraphs.Given two k-dicolourings of a digraph D, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for k=2 and for digraphs with maximum degree 5 or oriented planar graphs with maximum degree 6.A digraph is said to be k-mixing if there exists a transformation between any pair of k-dicolourings. We show that every digraph D is k-mixing for all k≥δmin∗(D)+2, generalizing a result due to Dyer et al. We also prove that every oriented graph G→ is k-mixing for all k≥δmax∗(G→)+1 and for all k≥δavg∗(G→)+1. Here δmin∗, δmax∗, and δavg∗ denote the min-degeneracy, the max-degeneracy, and the average-degeneracy respectively.We pose as a conjecture that, for every digraph D, the dicolouring graph of D on k≥δmin∗(D)+2 colours has diameter at most O(|V(D)|2). This is the analogue of Cereceda’s conjecture for digraphs. We generalize to digraphs two results supporting Cereceda’s conjecture. We first prove that the dicolouring graph of any digraph D on k≥2δmin∗(D)+2 colours has linear diameter, extending a result from Bousquet and Perarnau. We also prove that the analogue of Cereceda’s conjecture is true when k≥32(δmin∗(D)+1), which generalizes a result from Bousquet and Heinrich.Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on k≥2 colours is connected and has diameter at most 2n. We conjecture that every non 2-mixing oriented graph has maximum average degree at least 4, and we provide some support for this conjecture by proving it on the special case of 2-freezable oriented graphs. More generally, we show that every k-freezable oriented graph on n vertices must contain at least kn+k(k−2) arcs, and we give a family of k-freezable oriented graphs that reach this bound. In the general case, we prove as a partial result that every non 2-mixing oriented graph has maximum average degree at least 72.

On strong edge-coloring of graphs with maximum degree 5

A strong edge-coloring of a graph G is a proper edge-coloring such that any two edges with distance at most 2 receive different colors. The strong chromatic index of G, denoted by χs′(G), is the least possible number of colors in a strong edge-coloring of G. Erdős and Nešetřil conjectured that every graph G with maximum degree Δ(G) has χs′(G)≤54Δ(G)2−12Δ(G)+14 if Δ(G) is odd and χs′(G)≤54Δ(G)2 if Δ(G) is even. In this paper, we prove that if G is a graph with Δ(G)≤5 and maximum average degree less than 225, then χs′(G)≤29. Our result implies that Erdős’ conjecture holds for the case Δ(G)=5, if G has no subgraph with average degree at least 225.

Optimally reconfiguring list and correspondence colourings

The reconfiguration graph Ck(G) for the k-colourings of a graph G has a vertex for each proper k-colouring of G, and two vertices of Ck(G) are adjacent precisely when those k-colourings differ on a single vertex of G. Much work has focused on bounding the maximum value of diamCk(G) over all n-vertex graphs G. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if L is a list-assignment for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamCL(G)≤n(G)+μ(G). We also conjecture that if (L,H) is a correspondence cover for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamC(L,H)(G)≤n(G)+τ(G). (Here μ(G) and τ(G) denote the matching number and vertex cover number of G.) For every graph G, we give constructions showing that both conjectures are best possible, which also hints towards an exact form of Cereceda’s Conjecture for regular graphs. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) diamCL(G)≤n(G)+2μ(G) and diamC(L,H)(G)≤n(G)+2τ(G). Our second main result proves that both conjectured bounds hold, whenever all v satisfy |L(v)|≥2d(v)+1. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree. The full paper can also be found at arxiv.org/abs/2204.07928.

Amphibious Robot With Self-Rotating Paddle-Wheel Mechanism

Marine transportation is an important means of transportation, in which ships are in constant use and accidents are unavoidable. In this article, we developed an amphibious robot to help rescue people in an accidental environment that is too low or dangerous for rescuers to enter. Angled spoke paddling wheel (ASPW) makes it possible to drive on the water surface, while taking advantage of the angled spoke wheel (ASW). The ASPW changes the paddle force by rotating the paddle when the wheel rotates so that the total force is generated in the direction of the robot's movement. The paddle is optimized using the Taguchi method. The driving experiment was conducted in four environments: ground, water surface, transition between ground and water, and obstacle overcoming situation. The maximum driving speed was 0.47 m/s on the ground and 0.1 m/s on the water surface. The maximum average inclination degree in the transition situation was 25 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\circ }$</tex-math></inline-formula> from the ground to the water surface and 20 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\circ }$</tex-math></inline-formula> from the water surface to the ground. In addition, it was possible to overcome an obstacle with a maximum height of 41.3 mm, demonstrating that the proposed robot excels in amphibious driving, while possessing the ability to overcome the ASW level obstacle.

Mapping sparse signed graphs to (K2k,M) $({K}_{2k},M)$

AbstractA homomorphism of a signed graph to is a mapping of vertices and edges of to (respectively) vertices and edges of such that adjacencies, incidences, and the signs of closed walks are preserved. We observe in this work that, for , the ‐coloring problem of a given graph can be captured by homomorphism to from a signed bipartite graph that is built from . Here assigns a negative sign to the edges of a perfect matching and a positive sign to the rest. Motivated by these reformulations and in connection with results on 3‐colorings of planar graphs, such as Grötzsch's theorem, we prove that any signed graph with the maximum average degree strictly less than admits a homomorphism to . For , we show that the maximum average degree being strictly less than 3 would suffice for a signed graph to admit a homomorphism to . Both of these bounds are tight. We discuss applications of our work to signed planar graphs and its connection to the study of homomorphisms of 2‐edge‐colored graphs. Among a number of interesting questions that are left open, a notable one is a possible extension of Steinberg's conjecture for the class of signed bipartite planar graphs.

Weak degeneracy of graphs

AbstractMotivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every ‐degenerate graph is also weakly ‐degenerate. On the other hand, if is weakly ‐degenerate, then (and, moreover, the same bound holds for the list‐chromatic and even the DP‐chromatic number of ). It turns out that several upper bounds in graph coloring theory can be phrased in terms of weak degeneracy. For example, we show that planar graphs are weakly 4‐degenerate, which implies Thomassen's famous theorem that planar graphs are 5‐list‐colorable. We also prove a version of Brooks's theorem for weak degeneracy: a connected graph of maximum degree is weakly ‐degenerate unless . (By contrast, all ‐regular graphs have degeneracy .) We actually prove an even stronger result, namely that for every , there is such that if is a graph of weak degeneracy at least , then either contains a ‐clique or the maximum average degree of is at least . Finally, we show that graphs of maximum degree and either of girth at least 5 or of bounded chromatic number are weakly ‐degenerate, which is best possible up to the value of the implied constant.

Proper orientations and proper chromatic number

The proper orientation number χ→(G) of a graph G is the minimum k such that there exists an orientation of the edges of G with all vertex-outdegrees at most k and such that for any adjacent vertices, the outdegrees are different. Two major conjectures about the proper orientation number are resolved. First it is shown, that χ→(G) of any planar graph G is at most 14. Secondly, it is shown that for every graph, χ→(G) is at most ▪, where r=χ(G) is the usual chromatic number of the graph, and ▪ is the maximum average degree taken over all subgraphs of G. Several other related results are derived. Our proofs are based on a novel notion of fractional orientations.

Dense Overload Subgraph Induced by Cyber–Physical Attacks in Smart Grid

With the tight couplings between the power grid and communication networks, smart grid is vulnerable to cyber-physical attacks, which can inject imperceptive perturbations to cause cascading failures. Nevertheless, as a dynamic and evolving process, it remains challenging to precisely lock the susceptible components that initiate a cascading event. To address this issue, we formulate a bi-level optimization problem in the context of cyber-physical attacks to model the attacker-operator adversarial game and generate an optimal attack strategy for electricity vulnerability assessments. Specifically, we merge the electrical overloading, topological proximity and outage correlations to construct a dense overload subgraph, which is critical for cascading propagation, by making use of fractional program to capture the density of subgraph with maximum average degree. We then compute exact solutions by adopting multiple linearization techniques to convert this bi-level fractional program into a mixed-integer linear program. Numerical studies in IEEE different-scale power systems verify the effectiveness of our model in understanding the cascading risk for a smart grid.

Odd coloring of sparse graphs and planar graphs

An odd c-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petruševski and Škrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph G with mad(G)≤4c−4c+1 has an odd c-coloring for c≥4, and proved the conjecture for c∈{5,6}. In particular, planar graphs with girth at least 7 and 6 have an odd 5-coloring and an odd 6-coloring, respectively.We completely resolve Cranston's conjecture. For c≥7, we show that the conjecture is true, in a stronger form that was implicitly suggested by Cranston, but for c=4, we construct counterexamples, which all contain 5-cycles. On the other hand, we show that a graph G with mad(G)<229 and no induced 5-cycles has an odd 4-coloring. This implies that a planar graph with girth at least 11 has an odd 4-coloring. We also prove that a planar graph with girth at least 5 has an odd 6-coloring.

КИНО КАК КУЛЬТУРНАЯ ИНДУСТРИЯ: СОВРЕМЕННОЕ МИФОТВОРЧЕСТВО

This article analyzes the possibilities of using cinema to create and disseminate myths (political, social, cultural). At the same time, the role of cinema as a cultural industry is noted, where the creation of screen reality acts as a special type of myth-making, which is evaluated not only in terms of economic effect, but, above all, as a mechanism for creating new original images of social and therapeutic value. A film work is considered as a sign ensemble, while it is determined that the level of saturation with signs and the choice of sign systems are subject to the author's intention, accordingly, the question naturally arises: what is the author's intention “subject to”. At the same time, mass cinema builds its influence in such a way that the addressee receives the maximum possible emotional impression with guaranteed pleasure. Therefore, it is “tuned” to the “general” public with the maximum average degree of perception, with the most famous iconic signs. The paper provides an analysis of the mythology of Soviet cinema, Hollywood myth-making. According to the author of the article, the creators of modern myths, without ignoring the terms and concepts of "myth-making" and "mythologization", operate with a new concept - "mythotechnologies", which, depending on the goals, differ for cinema in that in the process a "new mythological image" or an existing one will be used.

On linear 2-arboricity of certain graphs

The linear 2-arboricity of a graph G is the least number of forests which decomposes E(G) and each forest is a collection of paths of length at most two. A graph has property Pk, if each subgraph H satisfies one of the three conditions: (i) δ(H)≤1; (ii) there exists xy∈E(H) with degH(x)+degH(y)≤k; (iii) H contains a 2-alternating cycle. In this paper, we give two edge-decompositions of graphs with property Pk. Using these decompositions, we give an upper bound for the linear 2-arboricity in terms of Pk. We also prove that every plane graph with no 12+-vertex incident with a gem at the center has property P13, and graphs with maximum average degree less than 6k−6k+3 have property Pk, where k≥5 is an integer.

Maximum average degree of list-edge-critical graphs and Vizing's conjecture

Vizing conjectured that χ ′ ℓ ( G )≤ Δ + 1 for all graphs. For a graph G and nonnegative integer k , we say G is a k -list-edge-critical graph if χ ′ ℓ ( G )&gt; k , but χ ′ ℓ ( G − e )≤ k for all e ∈ E ( G ) . We use known results for list-edge-critical graphs to verify Vizing’s conjecture for G with m a d ( G )&lt;( Δ + 3)/2 and Δ ≤ 9 .