On backbone coloring of graphs

Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G,H) is a mapping f: V(G)→{1,2,…,k} such that |f(u)−f(v)|≥2 if uv∈E(H) and |f(u)−f(v)|≥1 if uv∈E(G)\E(H). The backbone chromatic number of (G,H) is the smallest integer k such that (G,H) has a backbone-k-coloring. In this paper, we characterize the backbone chromatic number of Halin graphs G=T∪C with respect to given spanning trees T. Also we study the backbone coloring for other special graphs such as complete graphs, wheels, graphs with small maximum average degree, graphs with maximum degree 3, etc.     

Anti-forcing spectra of perfect matchings of graphs

Let M be a perfect matching of a graph G. The smallest number of edges whose removal to make M as the unique perfect matching in the resulting subgraph is called the anti-forcing number of M. The anti-forcing spectrum of G is the set of anti-forcing numbers of all perfect matchings in G, denoted by \(\hbox {Spec}_{af}(G)\). In this paper, we show that any finite set of positive integers can be the anti-forcing spectrum of a graph. We present two classes of hexagonal systems whose anti-forcing spectra are integer intervals. Finally, we show that determining the anti-forcing number of a perfect matching of a bipartite graph with maximum degree four is a NP-complete problem.     

Perfect matchings in paired domination vertex critical graphs

A vertex subset S of a graph G=(V,E) is a paired dominating set if every vertex of G is adjacent to some vertex in S and the subgraph induced by S contains a perfect matching. The paired domination number of G, denoted by γ _pr (G), is the minimum cardinality of a paired dominating set of?G. A?graph with no isolated vertex is called paired domination vertex critical, or briefly γ _pr -critical, if for any vertex v of G that is not adjacent to any vertex of degree one, γ _pr (G?v)<γ _pr (G). A?γ _pr -critical graph G is said to be k-γ _pr -critical if γ _pr (G)=k. In this paper, we firstly show that every 4-γ _pr -critical graph of even order has a perfect matching if it is K _1,5-free and every 4-γ _pr -critical graph of odd order is factor-critical if it is K _1,5-free. Secondly, we show that every 6-γ _pr -critical graph of even order has a perfect matching if it is K _1,4-free.     

Coupon coloring of some special graphs

Let G be a graph without isolated vertices. A k-coupon coloring of G is a k-coloring of G such that the neighborhood of every vertex of G contains vertices of all colors from \([k] =\{1, 2, \ldots , k\}\), which was recently introduced by Chen, Kim, Tait and Verstraete. The coupon coloring number \(\chi _c(G)\) of G is the maximum k for which a k-coupon coloring exists. In this paper, we mainly study the coupon coloring of some special classes of graphs. We determine the coupon coloring numbers of complete graphs, complete k-partite graphs, wheels, cycles, unicyclic graphs, bicyclic graphs and generalised \(\Theta \)-graphs.     

Trinque problem: covering complete graphs by plane degree-bounded hypergraphs

Let \(K_n\) be a complete graph drawn on the plane with every vertex incident to the infinite face. For any integers i and d, we define the (i, d)-Trinque Number of \(K_n\), denoted by \({\mathcal {T}}^d_{i}(K_n)\), as the smallest integer k such that there is an edge-covering of \(K_n\) by k “plane” hypergraphs of degree at most d and size of edge bounded by i. We compute this number for graphs (that is \(i=2\)) and gives some bounds for general hypergraphs.     

Dynamic matchings in left vertex weighted convex bipartite graphs

A left vertex weighted convex bipartite graph (LWCBG) is a bipartite graph \(G=(X,Y,E)\) in which the neighbors of each \(x\in X\) form an interval in \(Y\) where \(Y\) is linearly ordered, and each \(x\in X\) has an associated weight. This paper considers the problem of maintaining a maximum weight matching in a dynamic LWCBG. The graph is subject to the updates of vertex and edge insertions and deletions. Our dynamic algorithms maintain the update operations in \(O(\log ^2{|V|})\) amortized time per update, obtain the matching status of a vertex (whether it is matched) in constant worst-case time, and find the pair of a matched vertex (with which it is matched) in worst-case \(O(k)\) time, where \(k\) is not greater than the cardinality of the maximum weight matching. That achieves the same time bound as the best known solution for the problem of the unweighted version.     

An optimal square coloring of planar graphs

The square coloring of a graph is to color the vertices of a graph at distance at most 2 with different colors. In 1977, Wegner posed a conjecture on square coloring of planar graphs. The conjecture is still open. In this paper, we prove that Wegner’s conjecture is true for planar graphs with girth at least?6.     

Polynomial-time approximation algorithms for the coloring problem in some cases

We consider the coloring problem for hereditary graph classes, i.e. classes of simple unlabeled graphs closed under deletion of vertices. For the family of the hereditary classes of graphs defined by forbidden induced subgraphs with at most four vertices, there are three classes with an open complexity of the problem. For the problem and the open three cases, we present approximation polynomial-time algorithms with performance guarantees.     

On the max-weight edge coloring problem

We study the following generalization of the classical edge coloring problem: Given a weighted graph, find a partition of its edges into matchings (colors), each one of weight equal to the maximum weight of its edges, so that the total weight of the partition is minimized. We explore the frontier between polynomial and NP-hard variants of the problem, with respect to the class of the underlying graph, as well as the approximability of NP-hard variants. In particular, we present polynomial algorithms for bounded degree trees and star of chains, as well as an approximation algorithm for bipartite graphs of maximum degree at most twelve which beats the best known approximation ratios.     

Acyclic coloring of graphs with some girth restriction

A vertex coloring of a graph \(G\) is called acyclic if it is a proper vertex coloring such that every cycle \(C\) receives at least three colors. The acyclic chromatic number of \(G\) is the least number of colors in an acyclic coloring of \(G\). We prove that acyclic chromatic number of any graph \(G\) with maximum degree \(\Delta \ge 4\) and with girth at least \(4\Delta \) is at most \(12\Delta \).     

Adjacent vertex distinguishing edge coloring of IC-planar graphs

Journal of Combinatorial Optimization - The adjacent vertex distinguishing edge coloring of a graph G is a proper edge coloring in which each pair of adjacent vertices is assigned different color...     

Total coloring of planar graphs without short cycles

The total chromatic number of a graph \(G\), denoted by \(\chi ''(G)\), is the minimum number of colors needed to color the vertices and edges of \(G\) such that no two adjacent or incident elements get the same color. It is known that if a planar graph \(G\) has maximum degree \(\Delta (G)\ge 9\), then \(\chi ''(G)=\Delta (G)+1\). In this paper, it is proved that if \(G\) is a planar graph with \(\Delta (G)\ge 7\), and for each vertex \(v\), there is an integer \(k_v\in \{3,4,5,6,7,8\}\) such that there is no \(k_v\)-cycle which contains \(v\), then \(\chi ''(G)=\Delta (G)+1\).     

Algorithms for the maximum k-club problem in graphs

Given a simple undirected graph G, a k-club is a subset of vertices inducing a subgraph of diameter at most k. The maximum k-club problem (MkCP) is to find a k-club of maximum cardinality in G. These structures, originally introduced to model cohesive subgroups in social network analysis, are of interest in network-based data mining and clustering applications. The maximum k-club problem is NP-hard, moreover, determining whether a given k-club is maximal (by inclusion) is NP-hard as well. This paper first provides a sufficient condition for testing maximality of a given k-club. Then it proceeds to develop a variable neighborhood search (VNS) heuristic and an exact algorithm for MkCP that uses the VNS solution as a lower bound. Computational experiments with test instances available in the literature show that the proposed algorithms are very effective on sparse instances and outperform the existing methods on most dense graphs from the testbed.     

Equivalence of two conjectures on equitable coloring of graphs

A graph G is said to be equitably k-colorable if there exists a proper k-coloring of G such that the sizes of any two color classes differ by at most one. Let Δ(G) denote the maximum degree of a vertex in G. Two Brooks-type conjectures on equitable Δ(G)-colorability have been proposed in Chen and Yen (Discrete Math., 2011) and Kierstead and Kostochka (Combinatorica 30:201–216, 2010) independently. We prove the equivalence of these conjectures.     

Total coloring of planar graphs without adjacent short cycles

In the study of computer science, optimization, computation of Hessians matrix, graph coloring is an important tool. In this paper, we consider a classical coloring, total coloring. Let \(G=(V,E)\) be a graph. Total coloring is a coloring of \(V\cup {E}\) such that no two adjacent or incident elements (vertex/edge) receive the same color. Let G be a planar graph with \(\varDelta \ge 8\). We proved that if for every vertex \(v\in V\), there exists two integers \(i_v,j_v\in \{3,4,5,6,7\}\) such that v is not incident with adjacent \(i_v\)-cycles and \(j_v\)-cycles, then the total chromatic number of graph G is \(\varDelta +1\).     

The adjacent vertex distinguishing total coloring of planar graphs

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ≥14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.     

Acyclic edge coloring of planar graphs without 4-cycles

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a′(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiam?ik (Math. Slovaca 28:139–145, 1978) and later Alon, Sudakov and Zaks (J. Graph Theory 37:157–167, 2001) conjectured that a′(G)≤Δ+2 for any simple graph G with maximum degree Δ. In this paper, we confirm this conjecture for planar graphs G with Δ≠4 and without 4-cycles.     

Neighbor sum distinguishing list total coloring of subcubic graphs

Let \(G=(V, E)\) be a simple graph and denote the set of edges incident to a vertex v by E(v). The neighbor sum distinguishing (NSD) total choice number of G, denoted by \(\mathrm{ch}_{\Sigma }^{t}(G)\), is the smallest integer k such that, after assigning each \(z\in V\cup E\) a set L(z) of k real numbers, G has a total coloring \(\phi \) satisfying \(\phi (z)\in L(z)\) for each \(z\in V\cup E\) and \(\sum _{z\in E(u)\cup \{u\}}\phi (z)\ne \sum _{z\in E(v)\cup \{v\}}\phi (z)\) for each \(uv\in E\). In this paper, we propose some reducible configurations of NSD list total coloring for general graphs by applying the Combinatorial Nullstellensatz. As an application, we present that \(\mathrm{ch}^{t}_{\Sigma }(G)\le \Delta (G)+3\) for every subcubic graph G.