An application of the combinatorial Nullstellensatz to a graph labelling problem

An antimagic labelling of a graph G with m edges and n vertices is a bijection from the set of edges of G to the set of integers {1,…,m}, such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labelling. In N. Hartsfield and G. Ringle, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, Ringel has conjectured that every simple connected graph, other than K₂, is antimagic. In this article, we prove a special case of this conjecture. Namely, we prove that if G is a graph on n=p^k vertices, where p is an odd prime and k is a positive integer that admits a C_p‐factor, then it is antimagic. The case p=3 was proved in D. Hefetz, J Graph Theory 50 (2005), 263–272. Our main tool is the combinatorial Nullstellensatz [N. Alon, Combin Probab Comput 8(1–2) (1999), 7–29]. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 70–82, 2010.     

A new class of antimagic Cartesian product graphs

An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,…,m such that all n-vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [N. Hartsfield, G. Ringel, Pearls in Graph Theory, Academic Press, INC., Boston, 1990, pp. 108-109, Revised version, 1994] conjectured that every simple connected graph, except K₂, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In particular, by combining this result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in [Tao-Ming Wang, Toroidal grids are anti-magic, in: Proc. 11th Annual International Computing and Combinatorics Conference COCOON’2005, in: LNCS, vol. 3595, Springer, 2005, pp. 671-679], all Cartesian products of two or more regular graphs of positive degree can be proved to be antimagic.     

Anti‐magic graphs via the Combinatorial NullStellenSatz

An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1,…,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it has an antimagic labeling. In [ 10 ], Ringel conjectured that every simple connected graph, other than K₂, is antimagic. We prove several special cases and variants of this conjecture. Our main tool is the Combinatorial NullStellenSatz (cf. [ 1 ]). © 2005 Wiley Periodicals, Inc. J Graph Theory     

Regular bipartite graphs are antimagic

A labeling of a graph G is a bijection from E(G) to the set {1, 2,… |E(G)|}. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every connected graph other than K₂ is antimagic. In this article, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 173–182, 2009     

Antimagic orientations of even regular graphs

A labeling of a digraph D with m arcs is a bijection from the set of arcs of D to . A labeling of D is antimagic if no two vertices in D have the same vertex-sum, where the vertex-sum of a vertex for a labeling is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz [On antimagic directed graphs, J. Graph Theory 64 (2010) 219–232] initiated the study of antimagic labelings of digraphs, and conjectured that every connected graph admits an antimagic orientation, where an orientation D of a graph G is antimagic if D has an antimagic labeling. It remained unknown whether every disjoint union of cycles admits an antimagic orientation. In this article, we first answer this question in the positive by proving that every 2-regular graph has an antimagic orientation. We then show that for any integer , every connected, 2d-regular graph has an antimagic orientation. Our technique is new.     

A class of antimagic join graphs

A labeling f of a graph G is a bijection from its edge set E(G) to the set {1, 2, . . . , |E(G)|}, which is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. A graph G is antimagic if G has an f which is antimagic. Hartsfield and Ringel conjectured in 1990 that every connected graph other than K 2 is antimagic. In this paper, we show that if G 1 is an n-vertex graph with minimum degree at least r, and G 2 is an m-vertex graph with maximum degree at most 2r-1 (m ≥ n), then G1 ∨ G2 is antimagic.     

Construction of Antimagic Labeling for the Cartesian Product of Regular Graphs

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that that every connected graph, except K ₂, is antimagic. Recently, using completely separating systems, Phanalasy et al. showed that for each k 3 2, q 3 \binomk+12{k\geq 2,\,q\geq\binom{k+1}{2}} with k|2q, there exists an antimagic k-regular graph with q edges and p = 2q/k vertices. In this paper we prove constructively that certain families of Cartesian products of regular graphs are antimagic.     

On anti-magic labeling for graph products

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2,…,q} such that the vertex sums are pairwise distinct, where the vertex sum at one vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an anti-magic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K₂ are anti-magic. Recently, Alon et al. showed that this conjecture is true for dense graphs, i.e. it is true for p-vertex graphs with minimum degree Ω(logp). In this article, new classes of sparse anti-magic graphs are constructed through Cartesian products and lexicographic products.     

Antimagic orientation of Halin graphs

An antimagic labeling of a digraph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\dots ,m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum of a vertex is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. Hefetz, Mütze and Schwartz conjectured every connected undirected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that every Halin graph admits an antimagic orientation.     

Antimagic Properties of Graphs with Large Maximum Degree

An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In this article, we discuss antimagic properties of graphs that contain vertices of large degree. We also show that graphs with maximum degree at least are antimagic.     

Caterpillars with maximum degree 3 are antimagic

Let $G=(V,E)$ be a connected graph with $m$ edges. An antimagic labeling of $G$ is a one-to-one mapping from $E$ to $\{1,2,\dots ,m\}$ such that the vertex sum (i.e., sum of the labels assigned to edges incident to a vertex) for distinct vertices are different. A graph $G$ is called antimagic if $G$ has an antimagic labeling. It was conjectured by Hartsfield and Ringel that every tree other than $K_2$ is antimagic. The conjecture remains open though it was verified for trees with some constrains. Caterpillars are an important subclass of trees. This paper shows caterpillars with maximum degree 3 are antimagic, which gives an affirmative answer to an open problem of Lozano et al. (2019).     

Antimagic labeling of generalized pyramid graphs

An antimagic labeling of a graph withq edges is a bijection from the set of edges to the set of positive integers{1,2,...,q}such that all vertex weights are pairwise distinct,where the vertex weight of a vertex is the sum of the labels of all edges incident with that vertex.A graph is antimagic if it has an antimagic labeling.In this paper,we provide antimagic labelings for a family of generalized pyramid graphs.     

6‐Star‐Coloring of Subcubic Graphs

A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two adjacent vertices are assigned the same color) such that no path on four vertices is 2‐colored. The star chromatic number of G is the smallest integer k for which G admits a star coloring with k colors. In this paper, we prove that every subcubic graph is 6‐star‐colorable. Moreover, the upper bound 6 is best possible, based on the example constructed by Fertin, Raspaud, and Reed (J Graph Theory 47(3) (2004), 140–153).     

Antimagic Labelings of Join Graphs

An antimagic labeling of a graph with q edges is a bijection from the set of edges of the graph to the set of positive integers \({\{1, 2,\dots,q\}}\) such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. The join graph G + H of the graphs G and H is the graph with \({V(G + H) = V(G) \cup V(H)}\) and \({E(G + H) = E(G) \cup E(H) \cup \{uv : u \in V(G) {\rm and} v \in V(H)\}}\). The complete bipartite graph K _m,n is an example of join graphs and we give an antimagic labeling for \({K_{m,n}, n \geq 2m + 1}\). In this paper we also provide constructions of antimagic labelings of some complete multipartite graphs.     

The edge chromatic number of a directed/mixed multigraph

We consider colorings of the directed and undirected edges of a mixed multigraph G by an ordered set of colors. We color each undirected edge in one color and each directed edge in two colors, such that the color of the first half of a directed edge is smaller than the color of the second half. The colors used at the same vertex are all different. A bound for the minimum number of colors needed for such colorings is obtained. In the case where G has only directed edges, we provide a polynomal algorithm for coloring G with a minimum number of colors. An unsolved problem is formulated. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 267–273, 1999     

Extremal graphs in connectivity augmentation

Let A(n, k, t) denote the smallest integer e for which every k‐connected graph on n vertices can be made (k + t)‐connected by adding e new edges. We determine A(n, k, t) for all values of n, k, and t in the case of (directed and undirected) edge‐connectivity and also for directed vertex‐connectivity. For undirected vertex‐connectivity we determine A(n, k, 1) for all values of n and k. We also describe the graphs that attain the extremal values. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 179–193, 1999     

Antimagic labelling of vertex weighted graphs

Suppose G is a graph, k is a non‐negative integer. We say G is k‐antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted‐k‐antimagic if for any vertex weight function w: V→?, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well‐known conjecture asserts that every connected graph G≠K₂ is 0‐antimagic. On the other hand, there are connected graphs G≠K₂ which are not weighted‐1‐antimagic. It is unknown whether every connected graph G≠K₂ is weighted‐2‐antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted‐2‐antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted‐1‐antimagic. We also prove that every connected graph G≠K₂ on n vertices is weighted‐ ?3n/2?‐antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

Conservative graphs

A graph G with q edges is defined to be conservative if the edges of G can be oriented and distinctly numbered with the integers 1, 2,…, q so that at each vertex the sum of the numbers on the inwardly directed edges equals that on the outwardly directed edges. Several classes of graphs, including K_n, for n ≥4, and K_{2n, 2m}, for n, m ≥ 2, are shown to be conservative. It is proven that the dual of a planar graceful graph is conservative, and that the converse of this result is false.     

Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Approximate results for rainbow labelings

A simple graph \(G=(V,\,E)\) is said to be antimagic if there exists a bijection \(f{\text {:}}\,E\rightarrow [1,\,|E|]\) such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection \(f{\text {:}}\,V\rightarrow [1,\, |V|],\) such that \(\forall x,\,y\in V,\)

$$\begin{aligned} \sum _{x_i\in N(x)}f\left( x_i\right) \ne \sum _{x_j\in N(y)}f\left( x_j\right) . \end{aligned}$$

Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval \([1,\,2n+m-4]\) and, for trees with k inner vertices, in the interval \([1,\,m+k].\) In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree \(\Delta \) in the interval \([1,\,n+t(n-t)],\) where \( t=\min \{\Delta ,\,\lfloor n/2\rfloor \},\) and, for trees with k leaves, in the interval \([1,\, 3n-4k].\) In particular, all trees with \(n=2k\) vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.