NP completeness of finding the chromatic index of regular graphs
Abstract
We show that it is NP complete to determine whether it is possible to edge color a regular graph of degree k with k colors for any k ⩾ 3. As a by-product of this result, we obtain a new way to generate k-regular graphs which are k-edge colorable.
References (5)
- H Izbicki
An edge coloring problem
- I.J Holyer
The Computational Complexity of Graph Theory Problems
Cited by (181)
Hardness transitions and uniqueness of acyclic colouring
2024, Discrete Applied MathematicsFor , a -acyclic colouring of a graph is a function such that (i) for every edge of , and (ii) there is no cycle in bicoloured by . For , the problem -Acyclic Colourability takes a graph as input and asks whether admits a -acyclic colouring. Ochem (EuroComb 2005) proved that 3-Acyclic Colourability is NP-complete for bipartite graphs of maximum degree 4. Mondal et al. (2013) proved that 4-Acyclic Colourability is NP-complete for graphs of maximum degree five. We prove that for , -Acyclic Colourability is NP-complete for bipartite graphs of maximum degree , thereby generalising the NP-completeness result of Ochem, and adding bipartiteness to the NP-completeness result of Mondal et al.. In contrast, -Acyclic Colourability is polynomial-time solvable for graphs of maximum degree at most . Hence, for , the least integer such that -Acyclic Colourability in graphs of maximum degree is NP-complete, denoted by , satisfies . We prove that for , -Acyclic Colourability in -regular graphs is NP-complete if and only if . We also show that it is coNP-hard to check whether an input graph admits a unique -acyclic colouring up to colour swaps (resp. up to colour swaps and automorphisms).
Vertex-critical (P<inf>3</inf>+ℓP<inf>1</inf>)-free and vertex-critical (gem, co-gem)-free graphs
2024, Discrete Applied MathematicsA graph is -vertex-critical if but for all where denotes the chromatic number of . We show that there are only finitely many -vertex-critical -free graphs for all and all . Together with previous results, the only graphs for which it is unknown if there are an infinite number of -vertex-critical -free graphs is for all . We consider a restriction on the smallest open case, and show that there are only finitely many -vertex-critical (gem, co-gem)-free graphs for all , where gem. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of . This characterization allows us to give the complete list of all -vertex-critical (gem, co-gem)-free graphs for all .
Acyclic chromatic index of chordless graphs
2023, Discrete MathematicsAn acyclic edge coloring of a graph is a proper edge coloring with no bichromatic cycles. The acyclic chromatic index of a graph G denoted by , is the minimum integer k such that G has an acyclic edge coloring with k colors. It was conjectured by Fiamčík [13] that for any graph G with maximum degree Δ. Linear arboricity of a graph G, denoted by , is the minimum number of linear forests into which the edges of G can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. By a result of Basavaraju and Chandran [6], if G is chordless, then . Machado, de Figueiredo and Trotignon [23] proved that the chromatic index of a chordless graph is Δ when . We prove that for any chordless graph G, , when . Notice that this is an improvement over the result of Machado et al., since any acyclic edge coloring is also a proper edge coloring and we are using the same number of colors. As a byproduct, we prove that , when . To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado et al. [23] in case of chromatic index. This might be of independent interest.
A refinement on the structure of vertex-critical (P<inf>5</inf>, gem)-free graphs
2023, Theoretical Computer ScienceWe give a new, stronger proof that there are only finitely many k-vertex-critical (, gem)-free graphs for all k. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all 6- and 7-vertex-critical , gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the k-colourability of , gem)-free graphs for all k where the certificate is either a k-colouring or a -vertex-critical induced subgraph. Our complete lists for allow for the implementation of these algorithms for all .
A Markov chain on the solution space of edge colorings of bipartite graphs
2023, Discrete Applied MathematicsIn this paper, we exhibit an irreducible Markov chain on the edge -colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph and with the number of colors. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis–Hastings algorithm when the algorithm is applied on with the uniform distribution of all possible edge -colorings of . A special case of our results is the solution space of the possible completions of Latin rectangles.
Injective coloring of graphs revisited
2023, Discrete MathematicsAn open packing in a graph G is a set S of vertices in G such that no two vertices in S have a common neighbor in G. The injective chromatic number of G is the smallest number of colors assigned to vertices of G such that each color class is an open packing. Alternatively, the injective chromatic number of G is the chromatic number of the two-step graph of G, which is the graph with the same vertex set as G in which two vertices are adjacent if they have a common neighbor. The concept of injective coloring has been studied by many authors, while in the present paper we approach it from two novel perspectives, related to open packings and the two-step graph operation. We prove several general bounds on the injective chromatic number expressed in terms of the open packing number. In particular, we prove that holds for any connected graph G of order , size m, and the open packing number , and characterize the class of graphs attaining the bound. Regarding the well known bound , we describe the family of extremal graphs and prove that deciding when the equality holds (even for regular graphs) is NP-complete, solving an open problem from an earlier paper. Next, we consider the chromatic number of the two-step graph of a graph, and compare it with the clique number and the maximum degree of the graph. We present two large families of graphs in which equals the cardinality of a largest clique of the two-step graph of G. Finally, we consider classes of graphs that admit an injective coloring in which all color classes are maximal open packings. We give characterizations of three subclasses of these graphs among graphs with diameter 2, and find a partial characterization of hypercubes with this property.