Let $γ(G)$ denote the domination number of a graph $G$. A {\it Roman domination function} of a graph $G$ is a function $f: V\to\{0,1,2\}$ such that every vertex with 0 has a neighbor with 2. The {\it Roman domination number} $γ_R(G)$ is the minimum of $f(V(G))=Σ_{v\in V}f(v)$ over all such functions. Let $G\square H$ denote the Cartesian product of graphs $G$ and $H$. We prove that $γ(G)γ(H) \leq γ_R(G\square H)$ for all simple graphs $G$ and $H$, which is an improvement of $γ(G)γ(H) \leq 2γ(G\square H)$ given by Clark and Suen \cite{CS}, since $γ(G\square H)\leq γ_R(G\square H)\leq 2γ(G\square H)$.
For any graph $G=(V,E)$, a subset $S\subseteq V$ \emph{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 $γ(G \square H) \geq γ(G)γ(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we prove the conjecture.
A well-known conjecture of Vizing is that $γ(G \square H) \ge γ(G)γ(H)$ for any pair of graphs $G, H$, where $γ$ is the domination number and $G \square H$ is the Cartesian product of $G$ and $H$. Suen and Tarr, improving a result of Clark and Suen, showed $γ(G \square H) \ge \frac{1}{2}γ(G)γ(H) + \frac{1}{2}\min(γ(G),γ(H))$. We further improve their result by showing $γ(G \square H) \ge \frac{1}{2}γ(G)γ(H) + \frac{1}{2}\max(γ(G),γ(H)).$
For any graph $G=(V,E)$, a subset $S\subseteq V$ $dominates$ $G$ 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 $γ(G \square H) \geq γ(G)γ(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we prove that if $\left|G\right|\geq γ(G)γ(H)$ and $\left|H\right|\geq γ(G)γ(H)$, then the conjecture holds. This result quickly implies Vizing's conjecture for almost all pairs of graphs $G,H$ with $\left|G\right|\geq…
Let $γ(G)$ denote the domination number of graph $G$. Let $G$ and $H$ be graphs and $G\Box H$ their Cartesian product. For $h\in V(H)$ define $G_h=\{(g,h)\,|\,g\in V(G)\}$ and call this set a $G$-layer of $G\Box H$. We prove the following special case of Vizing's conjecture. Let $D$ be a dominating set of $G\Box H$. If there exist minimum dominating sets $D_1$ and $D_2$ of $G$ such that for every $h\in V(H)$, the projection of $D\cap G_h$ to $G$ is contained in $D_1$ or $D_2$, then $|D|\geq γ(G)γ(H)$.
We prove that if $M$ is a maximal $k$-edge-colorable subgraph of a multigraph $G$ and if $F = \{v \in V(G) : d_M(v) \leq k-μ(v)\}$, then $d_F(v) \leq d_M(v)$ for all $v \in F$. (When $G$ is a simple graph, the set $F$ is just the set of vertices having degree less than $k$ in $M$.) This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.
Let $G$ be a connected simple graph of order $n$ and let $Δ(G)$ and $χ'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $χ'(G)=Δ(G)$ or $Δ(G)+1$. Following this result, $G$ is called $Δ$-critical if $χ'(G)=Δ(G)+1$ and $χ'(G-e)=Δ(G)$ for every $e\in E(G)$. In 1968, Vizing conjectured that if $G$ is an $n$-vertex $Δ$-critical graph, then the independence number $α(G)\le n/2$. Furthermore, he conjectured that, in fact, $G$ has a 2-factor. Luo and Zhao showed that if $G$ is an $n$-vertex $Δ$-critical graph with $Δ(G)\ge n/2$, then $α(G)\le n/2$. More rece…
For any graph $G=(V,E)$, a subset $S\subseteq V$ \emph{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 $γ(G \square H) \geq γ(G)γ(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we define classes of graphs $\mathcal{A}_n$, for $n\geq 0$, so that every graph belongs to some such class, and $\mathcal{A}_0$ corresponds to class $A$ of Bartsalkin and German. We prove that for any graph $G$ in…
We show that if $G$ is a cograph, that is $P_4$-free, then for any graph $H$, $γ(G\square H)\geq γ(G)γ(H)$. By the characterization of cographs as a finite sequence of unions and joins of $K_1$, this result easily follows from that of Bartsalkin and German. However, the techniques used are new and may be useful to prove other results.
In 1964 Vizing proved that starting from any k-edge-coloring of a graph G one can reach, using only Kempe swaps, a ($Δ$ + 1)-edge-coloring of G where $Δ$ is the maximum degree of G. One year later he conjectured that one can also reach a $Δ$-edge-coloring of G if there exists one. Bonamy et. al proved that the conjecture is true for the case of triangle-free graphs. In this paper we prove the conjecture for all graphs.
Let $G$ be a simple graph, and let $Δ(G)$ and $χ'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $χ'(G)=Δ(G)$ or $Δ(G)+1$. We say $G$ is $Δ$-critical if $χ'(G)=Δ+1$ and $χ'(H)<χ'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $Δ$-critical graph, then $G$ has a 2-factor. Let $G$ be an $n$-vertex $Δ$-critical graph. It was proved that if $Δ(G)\ge n/2$, then $G$ has a 2-factor; and that if $Δ(G)\ge 2n/3+12$, then $G$ has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at le…
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 numer…
Let $G$ be a graph with maximum degree $Δ(G)$ and maximum multiplicity $μ(G)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $Δ(G)+μ(G)$. The distance between two edges $e$ and $f$ in $G$ is the length of a shortest path connecting an endvertex of $e$ and an endvertex of $f$. A distance-$t$ matching is a set of edges having pairwise distance at least $t$. Edwards et al. proposed the following conjecture: For any graph $G$, using the palette $\{1, \dots, Δ(G)+μ(G)\}$, any precoloring on a distance-$2$ matching can be extended to a proper edge co…
This paper has been withdrawn by the author. Peterson and Woodall previously proved that the list-edge-colouring conjecture holds for graphs without odd cycles of length 5 or longer. D. Peterson and D. R. Woodall, Edge-choosability in line-perfect multigraphs, Discrete Mathematics 202 (1999), 191-199. D. Peterson and D. R. Woodall, Erratum to "Edge-choosability in line-perfect multigraphs", Discrete Mathematics 260 (2003), 323-326.
Vizing's conjecture (open since 1968) relates the sizes of dominating sets in two graphs to the size of a dominating set in their Cartesian product graph. In this paper, we formulate Vizing's conjecture itself as a Positivstellensatz existence question. In particular, we encode the conjecture as an ideal/polynomial pair such that the polynomial is nonnegative if and only if the conjecture is true. We demonstrate how to use semidefinite optimization techniques to computationally obtain numeric sum-of-squares certificates, and then show how to transform these numeric certificates into symbolic c…
Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $γ(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{0, 1, . . ., k\}$ is called a \emph{$\{k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{k\}$-dominating function is the sum of its values over all the vertices. The $\{k\}$-domination number of $G$, $γ_{\{k\}}(G)$, is defined to be t…
The total chromatic number conjecture which has appeared in a few hundred articles and in numerous books thus far is now one of the classic mathematical unsolved problems. It appears that many authors coincidentally have attributed it to Professor M. Behzad and/or to Professor V. G. Vizing. Eventually after four decades, Professor A. Soifer investigated the origin of this conjecture; published his findings in The Mathematical Coloring Book - 2009; and stated that, "In my opinion this unquestionably merits the joint credit to Vizing and Behzad." After checking all the arguments presented and th…
Fix a palette $\mathcal K$ of $Δ+1$ colours, a graph with maximum degree $Δ$, and a subset $M$ of the edge set with minimum distance between edges at least $9$. If the edges of $M$ are arbitrarily precoloured from $\mathcal K$, then there is guaranteed to be a proper edge-colouring using only colours from $\mathcal K$ that extends the precolouring on $M$ to the entire graph. This result is a first general precolouring extension form of Vizing's theorem, and it proves a conjecture of Albertson and Moore under a slightly stronger distance requirement. We also show that the condition on the dista…
The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge-colorings for graphings. A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence theory and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge-coloring with $d + O(\sqrt{d})$ colors; furthermo…
An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Goldberg conjecture for multigraphs. The results obtained in the paper support our con…
Recently the notion of $k$-rainbow total domination was introduced for a graph $G$, motivated by a desire to reduce the problem of computing the total domination number of the generalized prism $G \Box K_k$ to an integer labeling problem on $G$. In this paper we further demonstrate usefulness of the labeling approach, presenting bounds on the rainbow total domination number in terms of the total domination number, the rainbow domination number and the rainbow total domination number, as well as the usual domination number, where the latter presents a generalization of a result by Goddard and H…
The only open case of Vizing's conjecture that every planar graph with $Δ\geq 6$ is a class 1 graph is $Δ= 6$. We give a short proof of the following statement: there is no 6-critical plane graph $G$, such that every vertex of $G$ is incident to at most three 3-faces. A stronger statement without restriction to critical graphs is stated in \cite{Wang_Xu_2013}. However, the proof given there works only for critical graphs. Furthermore, we show that every 5-critical plane graph has a 3-face which is adjacent to a $k$-face $(k\in \{3,4\})$. For $Δ= 5$ our result gives insights into the structure …
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-c…
Signed list edge coloring in graphs of bounded treewidth
Yan’an University（延安大学）；School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；School of Mathematics and Statistics & Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学数学与统计学院与分析数学及应用重点实验室（教育部））；School of Marine Science and Technology, Northwestern Polytechnical University（西北工业大学海洋科学与技术学院）
Edwards, van den Heuvel, Kang, and Sereni conjectured the following strengthening of Vizing's Theorem: let $G$ be a simple graph, and let $K = Δ(G) + 1$. For any matching $M$ in $G$ and any precoloring of the edges in $M$ using the colors $\{1, \ldots, K\}$, there is some proper $K$-edge-coloring of $G$ extending the given precoloring. We give an infinite family of counterexamples to this conjecture, and prove a weaker version of the conjecture proposed in the same work.
We conjecture that any graph $G$ with treewidth~$k$ and maximum degree $Δ(G)\geq k + \sqrt{k}$ satisfies $χ'(G)=Δ(G)$. In support of the conjecture we prove its fractional version. We also show that any graph $G$ with treewidth~$k\geq 4$ and maximum degree $2k-1$ satisfies $χ'(G)=Δ(G)$, improving an old result of Vizing.
Given a graph $G$, denote by $Δ$, $\bar{d}$ and $χ^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$Δ$-critical} if $χ^\prime(G)=Δ+1$ and $χ^\prime(H)\leΔ$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is edge-$Δ$-critical, then $\bar{d}\geq Δ-1+ \frac{3}{n}$. We show that $$ \begin{displaystyle} \avd \ge \begin{cases} 0.69241\D-0.15658 \quad\,\: \mbox{ if } Δ\geq 66, 0.69392\D-0.20642\quad\;\,\mbox{ if } Δ=65, \mbox{ and } 0.68706\D+0.19815\quad\! \quad\mbox{if } 56\leq Δ\leq64. …
There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conje…
Let $G=(V(G), E(G))$ be a multigraph with maximum degree $Δ(G)$, chromatic index $χ'(G)$ and total chromatic number $χ''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $χ''(G)\leq Δ(G)+μ(G) +1$ for a multigraph $G$, where $μ(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $χ''(G)=χ'(G)$ if $χ'(G)\geq Δ(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $χ''(G)=χ'(G)$ if $χ'(G)\geq \max\{ Δ(G)+2, |V(G)|+1\}$ in this note. Consequently, $χ'…
A simple graph $G$ is \emph{overfull} if $|E(G)|>Δ\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $χ'(G)>Δ$. The \emph{core} of a graph, denoted $G_Δ$, is the subgraph induced by its vertices of degree $Δ$. Vizing's Adjacency Lemma implies that if $χ'(G)>Δ$, then $G_Δ$ contains cycles. Hilton and Zhao conjectured that if $G_Δ$ has maximum degree 2 and $Δ\ge 4$, then $χ'(G)>Δ$ precisely when $G$ is overfull. We prove this conjecture for the case $Δ=4$.
For graphs $G,H$ it is possible to add $(|V(G)|-γ(G))(|V(H)|-γ(H))$ edges to the Cartesian product $G\mathbin{\square}H$ such that a minimal dominating set $D$ of size $γ(G)γ(H)$ emerges. We hypothesize that $D$ is also a minimum dominating set for the resulting graph and show that this implies Vizing's conjecture.
\textit{Total Coloring} of a graph is a major coloring problem in combinatorial mathematics, introduced in the early $1960$s. A \textit{total coloring} of a graph $G$ is a map $f:V(G) \cup E(G) \rightarrow \mathcal{K}$, where $\mathcal{K}$ is a set of colors, satisfying the following three conditions: 1. $f(u) \neq f(v)$ for any two adjacent vertices $u, v \in V(G)$; 2. $f(e) \neq f(e')$ for any two adjacent edges $e, e' \in E(G)$; and 3. $f(v) \neq f(e)$ for any vertex $v \in V(G)$ and any edge $e \in E(G)$ that is incident to the same vertex $v$. The \textit{total chromatic number}, $χ''(G)$…
Let $G$ be a simple graph with maximum degree $Δ(G)$ and chromatic index $χ'(G)$. A classic result of Vizing indicates that either $χ'(G )=Δ(G)$ or $χ'(G )=Δ(G)+1$. The graph $G$ is called $Δ$-critical if $G$ is connected, $χ'(G )=Δ(G)+1$ and for any $e\in E(G)$, $χ'(G-e)=Δ(G)$. Let $G$ be an $n$-vertex $Δ$-critical graph. Vizing conjectured that $α(G)$, the independence number of $G$, is at most $\frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $α(G)<\frac{3n}{5}$. We show that for any given $\varepsilon\in (0,1)$, there exist positive constants $d_0(\vareps…
The total chromatic number, $χ''(G)$ is the minimum number of colors which need to be assigned to obtain a total coloring of the graph $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing that for any graph, $χ''(G) \leq Δ(G)+2 $, where $Δ(G)$ represents the maximum degree of $G$. In this paper we obtained the total chromatic number for some classes of four regular circulant graphs.
In this paper we study the the average order of dominating sets in a graph, $\operatorname{avd}(G)$. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown (2021) conjectured that for all graphs $G$ of order $n$ without isolated vertices, $\operatorname{avd}(G) \leq 2n/3$. Recently, Erey (2021) proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have $\operatorname{avd}(G) = 2n/3$. We also use our bounds to prove the average version of Vizing's Conjecture.
Let $G$ be a simple graph with maximum degree $Δ$. We call $G$ \emph{overfull} if $|E(G)|>Δ\lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_Δ$, is the subgraph of $G$ induced by its vertices of degree $Δ$. A classic result of Vizing shows that $χ'(G)$, the chromatic index of $G$, is either $Δ$ or $Δ+1$. It is NP-complete to determine the chromatic index for a general graph. However, if $G$ is overfull then $χ'(G)=Δ+1$. Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $Δ\ge 3$ and $Δ(G_Δ)\le 2$, then $χ'(G)=Δ+1$ if and only if $G$ is overfull or $G=P^…
In this paper, we show that every $(2P_2,K_4)$-free graph is 4-colorable. The bound is attained by the five-wheel and the complement of the seven-cycle. This answers an open question by Wagon \cite{Wa80} in the 1980s. Our result can also be viewed as a result in the study of the Vizing bound for graph classes. A major open problem in the study of computational complexity of graph coloring is whether coloring can be solved in polynomial time for $(4P_1,C_4)$-free graphs. Lozin and Malyshev \cite{LM17} conjecture that the answer is yes. As an application of our main result, we provide the first …
Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq Δ(G)$, while by Vizing's theorem, no more than $k=Δ(G)+1$ is necessary for constructing such $c$. On the course of investigating irregularities in graphs, it has been moreover conjectured that only slightly larger $k$, i.e., $k=Δ(G)+2$ enables enforcing additional strong feature of $c$, namely that it attributes distinct sums of incident colours to adjacent vertices in $G$ if only this graph has no isolated edges and is not isomorphic to $C_5$. We prove the conjecture is valid for planar graphs of…
Given a graph $G$ possibly with multiple edges but no loops, denote by $Δ$ the {\it maximum degree}, $μ$ the {\it multiplicity}, $χ'$ the {\it chromatic index} and $χ_f'$ the {\it fractional chromatic index} of $G$, respectively. It is known that $Δ\le χ_f' \le χ' \le Δ+ μ$, where the upper bound is a classic result of Vizing. While deciding the exact value of $χ'$ is a classic NP-complete problem, the computing of $χ_f'$ is in polynomial time. In fact, it is shown that if $χ_f' > Δ$ then $χ_f'= \max \frac{|E(H)|}{\lfloor |V(H)|/2\rfloor}$, where the maximality is over all induced subgraphs $H…
Let $G$ be a simple graph with maximum degree $Δ$. A classic result of Vizing shows that $χ'(G)$, the chromatic index of $G$, is either $Δ$ or $Δ+1$. We say $G$ is of \emph{Class 1} if $χ'(G)=Δ$, and is of \emph{Class 2} otherwise. A graph $G$ is \emph{$Δ$-critical} if $χ'(G)=Δ+1$ and $χ'(H)<Δ+1$ for every proper subgraph $H$ of $G$, and is \emph{overfull} if $|E(G)|>Δ\lfloor (|V(G)|-1)/2 \rfloor$. Clearly, overfull graphs are Class 2. Hilton and Zhao in 1997 conjectured that if $G$ is obtained from an $n$-vertex $Δ$-regular Class 1 graph with maximum degree greater than $n/3$ by splitting a v…
Let $G$ be a $k$-connected graph on $n$ vertices. Hippchen's Conjecture states that two longest paths in $G$ share at least $k$ vertices. Gutiérrez recently proved the conjecture when $k\leq 4$ or $k\geq \frac{n-2}{3}$. We improve upon both results; namely, we show that two longest paths in $G$ share at least $k$ vertices when $k=5$ or $k\geq \frac{n+2}{5}$. This completely resolves two conjectures of Gutiérrez in the affirmative.
A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $γ(G)$ is the size of a minimum dominating set in $G$. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $γ(G) γ(H) \leq γ(G \Box H)$, and Clark and Suen (2000) proved that $γ(G) γ(H) \leq 2γ(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ t…
A global offensive alliance in a graph $G$ is a set $S$ of vertices with the property that every vertex not belonging to $S$ has at least one more neighbor in $S$ than it has outside of $S$. The global offensive alliance number of $G$, $γ_o(G)$, is the minimum cardinality of a global offensive alliance in $G$. A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex not belonging to $S$ has at least one neighbor in $S$. The domination number of $G$, $γ(G)$, is the minimum cardinality of a dominating set of $G$. In this work we obtain closed formulas for the global offen…
