Vizing's Conjecture Research Articles

A tight upper bound on the average order of dominating sets of a graph

AbstractIn this paper we study the average order of dominating sets in a graph, . Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs of order without isolated vertices, . Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have . We also use our bounds to prove an average version of Vizing's conjecture.

Sum-of-squares certificates for Vizing's conjecture via determining Gröbner bases

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar, Krenn, Margulies and Wiegele used the graph class G of all graphs with nG vertices and domination number kG and reformulated Vizing's conjecture as the problem that for all graph classes G and H the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of kG, nG, kH, and nH.In this paper, we consider their approach for kG=kH=1. For this case we are able to derive the unique reduced Gröbner basis of the Vizing ideal. Based on this, we deduce the minimum degree (nG+nH−1)/2 of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes G and H with nG+nH−1=d for general d, which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes G and H with kG=kH=1 and nG+nH≤15.

Total coloring conjecture on certain classes of product graphs

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G , denoted by χ ″( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G , Δ ( G )+1 ≤ χ ″( G )≤ Δ ( G )+2 , where Δ ( G ) is the maximum degree of G . In this paper, we prove the Behzad and Vizing conjecture for Indu - Bala product graph, Skew and Converse Skew product graph, Cover product graph, Clique cover product graph and Comb product graph.

Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional Grids

In this paper, we study a variant of graph domination known as $(t, r)$ broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour in 2015. In this variant, each broadcast provides $t-d$ reception to each vertex a distance $d < t$ from the broadcast. If $d \ge t$ then no reception is provided. A vertex is considered dominated if it receives $r$ total reception from all broadcasts. Our main results provide some upper and lower bounds on the density of a $(t, r)$ dominating pattern of an infinite grid, as well as methods of computing them. Also, when $r \ge 2$ we describe a family of counterexamples to a generalization of Vizing's Conjecture to $(t,r)$ broadcast domination.

Signed planar graphs with Δ ≥ 8 are Δ-edge-colorable

A well-known theorem due to Vizing states that every graph with maximum degree Δ is Δ- or (Δ+1)-edge-colorable. Recently, Behr extended the concept of edge coloring in a natural way to signed graphs. He also proved that an analogue of Vizing's Theorem holds for all signed graphs. Adopting Behr's definition, Zhang et al. proved that a signed planar graph G with maximum degree Δ is Δ-edge-colorable if either Δ≥10 or Δ∈{8,9} and G contains no adjacent triangles. They also proposed the conjecture that every signed planar graph with Δ≥6 is Δ-edge-colorable, as a generalization of Vizing's Planar Graph Conjecture. In this paper, we prove that every signed planar graph with Δ≥8 is Δ-edge-colorable.

The average degree of edge chromatic critical graphs with maximum degree seven

AbstractIn this paper, by developing several new adjacency lemmas about a path on four or five vertices, we show that the average degree of 7‐critical graphs is at least 6. It implies Vizing's planar graph conjecture for planar graphs with maximum degree 7 and its extension to graphs embeddable in a surface with nonnegative Euler characteristic due to Sanders and Zhao, and Zhang.

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 )> 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 )<( Δ + 3)/2 and Δ ≤ 9 .

On construction for trees making the equality hold in Vizing's conjecture

AbstractIn 1968, Vizing gave a conjecture: for any graphs and , which is now still open. In the text book “Domination in Graph: Advanced Topices” edited by Haynes et al., they listed such a question: is there a structural characterization of the graphs such that there exists a graph with ? In this paper, we restrict to be a tree and prove that: for any tree , there exists a nonempty graph such that if and only if . Based on this, we obtain a construction of the tree class is a nontrivial tree and there exists a nonempty graph such that .

Graph Theory Algorithms of Hamiltonian Cycle from Quasi‐Spanning Tree and Domination Based on Vizing Conjecture

In this study, from a tree with a quasi‐spanning face, the algorithm will route Hamiltonian cycles. Goodey pioneered the idea of holding facing 4 to 6 sides of a graph concurrently. Similarly, in the three connected cubic planar graphs with two‐colored faces, the vertex is incident to one blue and two red faces. As a result, all red‐colored faces must gain 4 to 6 sides, while all obscure‐colored faces must consume 3 to 5 sides. The proposed routing approach reduces the constriction of all vertex colors and the suitable quasi‐spanning tree of faces. The presented algorithm demonstrates that the spanning tree parity will determine the arbitrary face based on an even degree. As a result, when the Lemmas 1 and 2 theorems are compared, the greedy routing method of Hamiltonian cycle faces generates valuable output from a quasi‐spanning tree. In graph idea, a dominating set for a graph S = (V, E) is a subset D of V. The range of vertices in the smallest dominating set for S is the domination number (S). Vizing’s conjecture from 1968 proves that the Cartesian fabricated from graphs domination variety is at least as big as their domination numbers production. Proceeding this work, the Vizing’s conjecture states that for each pair of graphs S, L.

Bounding the k-rainbow total domination number

Let [k]={1,2,…,k}, where the elements of [k] are called colors. Supposing f is a function which assigns a subset of [k] to each vertex of a graph G, if each vertex which is assigned the empty set has all k colors in its neighborhood, then f is called a k-rainbow dominating function for G. If f satisfies the additional condition that for every v∈V(G) such that f(v)={i} for some i∈[k] there exists u∈NG(v) such that i∈f(u), then f is called a k-rainbow total dominating function. The corresponding invariant γkrt(G), which is the minimum sum of numbers of assigned colors over all vertices of G, is called the k-rainbow total domination number of G. We present bounds on this domination invariant in terms of domination, total domination, and k-rainbow (total) domination. We compute γkrt(Ka,b), and use this result in a number of places. We establish that for k≥3, the following bounds are tight: γ(k−1)rt(G)≤γkrt(G)≤kk−1γ(k−1)rt(G); we prove similar bounds related to rainbow domination and total domination. For a graph G, its ordinary domination number γ(G), and k≥2, we prove that γkrt(G)≥2kk+1γ(G), which presents a generalization of a result for the case k=2 by Goddard and Henning (2018); we then consider some implications of this result. We conclude the paper with Vizing-like conjectures for both rainbow domination and rainbow total domination, and relate the later one with the original Vizing conjecture.

Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares

Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question. In particular, we select classes of graphs according to their number of vertices and their domination number and encode the conjecture as an ideal/polynomial pair such that the polynomial is non-negative on the variety associated with the ideal if and only if the conjecture is true for this graph class. Using semidefinite programming we obtain numeric sum-of-squares certificates, which we then manage to transform into symbolic certificates confirming non-negativity of our polynomials. Specifically, we obtain exact low-degree sparse sum-of-squares certificates for particular classes of graphs.The obtained certificates allow generalizations for larger graph classes. Besides computational verification of these more general certificates, we also present theoretical proofs as well as conjectures and questions for further investigations.

A Note on Edge-Group Choosability of Planar Graphs without 5-Cycles

This paper is devoted to a study of the concept of edge-group choosability of graphs. We say that G is edge- k -group choosable if its line graph is k -group choosable. In this paper, we study an edge-group choosability version of Vizing conjecture for planar graphs without 5-cycles and for planar graphs without noninduced 5-cycles (2010 Mathematics Subject Classification: 05C15, 05C20).

A new framework to approach Vizing's conjecture

A new framework to approach Vizing's conjecture

An Improvement in the Two-packing Bound Related to Vizing's Conjecture

Vizing's conjecture states that the domination number of the Cartesian product of graphs is at least the product of the domination numbers of the two factor graphs. In this note we improve the recent bound of Breŝar by applying a technique of Zerbib to show that for any graphs G and H, γ(G x H)≥ γ (G) 2/3(γ(H)-ρ(H)+1), where γ is the domination number, ρ is the 2-packing number, and x is the Cartesian product.

Some improved inequalities related to Vizing's conjecture

Let γ(G) be the domination number of a simple graph G and G□H be the Cartesian product of two simple graphs G and H. A function f:V(G)→{0,1,2} is a Roman dominating function (RDF) if for each vertex u∈V0,NG(u)∩V2≠∅, where Vi={u∈V(G):f(u)=i}. The Roman domination number γR(G) is the minimum weight f(V(G))=∑u∈V(G)f(u) among all RDFs of G. Vizing conjectured in 1963 that γ(G□H)≥γ(G)γ(H) for any graphs G and H. To this day, this conjecture remains open. In this paper, we show that for each pair of simple graphs G and H, γ(G□H)≥14γR(G)γR(H). This means that Vizing's conjecture holds for any pair of Roman graphs G and H. Moreover, we prove γR(G□H)≥γ(G)γ(H)+12min⁡{γ(G),γ(H)} if G or H is nonempty, which is a slight improvement of γR(G□H)≥γ(G)γ(H) obtained by Wu in 2013 [22].

Total Colorings of Product Graphs

A total coloring of a graph is an assignment of colors to all the elements of the graph in such a way that no two adjacent or incident elements receive the same color. In this paper, we prove Behzad–Vizing conjecture for product graphs. In particular, we obtain the tight bound for certain classes of graphs.

Face-Degree Bounds for Planar Critical Graphs

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs, based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist.For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs.

A Class of Graphs Approaching Vizing's Conjecture

For any graph G=(V,E), a subset S of V dominates G if all vertices are contained in the closed neighborhood of S, that is N[S]=V. The minimum cardinality over all such S is called the domination number, written γ(G). In 1963, V.G. Vizing conjectured that γ(G □ H) ≥ γ(G)γ(H) where □ stands for the Cartesian product of graphs. In this note, we define classes of graphs An, for n≥0, so that every graph belongs to some such class, and A0 corresponds to class A of Bartsalkin and German. We prove that for any graph G in class A1, γ(G□H)≥ [γ(G)-√(γ(G))]γ(H).

The spectral radius of edge chromatic critical graphs

A connected graph G with maximum degree Δ and edge chromatic number χ′(G)=Δ+1 is called Δ-critical if χ′(G−e)=Δ for every edge e of G. In this paper, we consider two weaker versions of Vizing's conjecture, which concern the spectral radius ρ(G) and the signless Laplacian spectral radius μ(G) of G. We obtain some lower bounds for ρ(G) and μ(G), and present some cases where the conjectures are true. Finally, several open problems are also proposed.

Finding Δ(Σ) for a Surface Σ of Characteristic −4

For each surface Σ, we define max G is a class two graph of maximum degree that can be embedded in . Hence, Vizing's Planar Graph Conjecture can be restated as if Σ is a sphere. In this article, by applying some newly obtained adjacency lemmas, we show that if Σ is a surface of characteristic . Until now, all known satisfy . This is the first case where .