Determining chromatic index of cubic graph with the use of explainable classifiers: A comparative study

We consider constrained proper edge colorings of the following type: Given a positive integer $j$ and a family $\mathcal{F}$ of connected graphs on three or more vertices, we require that the subgraph formed by the union of any $j$ color classes has no copy of any member of $\mathcal{F}$. This generalizes some well-known types of colorings such as acyclic edge colorings, distance-2 edge colorings, low treewidth edge colorings, etc. For such a generalization of restricted colorings, we obtain an upper bound of $O(d^{\max(\theta,1)})$ on the minimum number of colors used in such a coloring. Here $d$ refers to the maximum degree of the graph, and $\theta$ is a parameter defined by $\theta=\theta(j,\mathcal{F})=\mathit{SUP}_{H\in\mathcal{F}}\frac{(|V(H)|-2)}{(|E(H)|-j)}$, where SUP stands for the supremum. Our proof is based on probabilistic arguments. In particular, we obtain $O(d)$ upper bounds for proper edge colorings with various interesting restrictions placed on the union of color classes. For example, we obtain $O(d)$ upper bounds on edge colorings with restrictions such as (i) the union of any three color classes should be an outerplanar graph, (ii) the union of any four color classes should have treewidth at most 2, (iii) the union of any five color classes should be planar, (iv) the union of any 16 color classes should be 5-degenerate, etc. We also consider generalizations where we require simultaneously for several pairs $(j_i,\mathcal{F}_i)$ ($i=1,\dots,s$) that the union of any $j_i$ color classes has no copy of any member of $\mathcal{F}_i$ and obtain upper bounds on the corresponding chromatic indices. As a corollary, we obtain that each of the four restrictions above can be satisfied simultaneously using $O(d)$ colors. Some ways of improving the bounds are sketched. Also, if we drop the requirement that the edge coloring be proper, then an $O(d^{\theta})$ upper bound on the chromatic index is established. Further, the stated upper bounds are also bounds for the list analogues of these edge colorings.

Coloring the cliques of line graphs

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.

We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.

A vertex coloring of a given graph is conflict‐free if the closed neighborhood of every vertex contains a unique color (i.e., a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the conflict‐free chromatic number of , denoted . What is the maximum possible conflict‐free chromatic number of a graph with a given maximum degree ? Trivially, , but it is far from optimal—due to results of Glebov, Szabó, and Tardos, and of Bhyravarapu, Kalyanasundaram, and Mathew, the answer is known to be . We show that the answer to the same question in the class of line graphs is —it follows that the extremal value of the conflict‐free chromatic index among graphs with maximum degree is much smaller than the one for conflict‐free chromatic number. The same result for is also provided in the class of near regular graphs, that is, graphs with minimum degree .

The sum of a coloring is the sum of the colors assigned to the vertices (assuming that the colors are positive integers). The sum ? (G) of graph G is the smallest sum that can be achieved by a proper vertex coloring of G. The chromatic strength s(G) of G is the minimum number of colors that is required by a coloring with sum ? (G). For every k, we determine the complexity of the question Is s(G) ? k?: it is coNP-complete for k = 2 and ?2p-complete for every fixed k ? 3. We also study the complexity of the edge coloring version of the problem, with analogous definitions for the edge sum ??(G) and the chromatic edge strength s?(G). We show that for every k ? 3, it is ?2p-complete to decide whether s?(G) ? k. As a first step of the proof, we present graphs for every r ? 3 with chromatic index r and edge strength r + 1. For some values of r, such graphs have not been known before.

A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in an acyclic edge colouring of G. In this paper, we discuss the acyclic edge colouring of middle, central, total and line graphs of prime related star graph families. Also exact values of acyclic chromatic indices of such graphs are derived and some of their structural properties are discussed.

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e. with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so far has focused on coloring graphs of maximum degree Delta with at most Delta+1 colors (or Delta colors when some simple obstructions are forbidden). When Delta is sufficiently large and c >= Delta-k_Delta+1, for some integer k_Delta ~~ sqrt{Delta}-2, we give a distributed algorithm that given a c-colorable graph G of maximum degree Delta, finds a c-coloring of G in min{O((log Delta)^{13/12}log n), 2^{O(log Delta+sqrt{log log n})}} rounds, with high probability. The lower bound Delta-k_Delta+1 is best possible in the sense that for infinitely many values of Delta, we prove that when chi(G) = Delta-k_Delta deciding whether chi(G) <= c is in P, while Embden-Weinert et al. proved that for c <= Delta-k_Delta-1, the same problem is NP-complete. Note that the sequential and distributed thresholds differ by one. Our first result covers the case where the chromatic number of the graph ranges between Delta-sqrt{Delta} and Delta+1. Our second result covers a larger range, but gives a weaker bound on the number of colors: For any sufficiently large Delta, and Omega(log Delta) 0, with a randomized algorithm running in O(log n/log log n) rounds with high probability.

The k-2-distance coloring of a graph G is a mapping c: V (G) →{ 1, 2, ··· ,k } such that for every pair of u, v satisfying 0 <d G(u, v) ≤ 2, c(u) � c(v). The minimum number of colors in 2-distance coloring of G is its 2-distance chromatic number, denoted by χ2(G). In this paper, we prove that every planar graph without 3, 4, 8-cycles and Δ ≥ 14 is (Δ + 5)-2-distance colorable.

On sum coloring of graphs

A vertex colouring of a graph is asymmetric if it is preserved only by the identity automorphism. The minimum number of colours needed for an asymmetric colouring of a graph $G$ is called the asymmetric colouring number or distinguishing number $D(G)$ of $G$. It is well known that $D(G)$ is closely related to the least number of vertices moved by any non-identity automorphism, the so-called motion $m(G)$ of $G$. Large motion is usually correlated with small $D(G)$. Recently, Babai posed the question whether there exists a function $f(d)$ such that every connected, countable graph $G$ with maximum degree $\Delta(G)\leq d$ and motion $m(G)&gt;f(d)$ has an asymmetric $2$-colouring, with at most finitely many exceptions for every degree.&#x0D; We prove the following result: if $G$ is a connected, countable graph of maximum degree at most 4, without an induced claw $K_{1,3}$, then $D(G)= 2$ whenever $m(G)&gt;2$, with three exceptional small graphs. This answers the question of Babai for $d=4$ in the class of~claw-free graphs.

In this paper we introduce an approach for solving the graph coloring problem. The problem is a NP-complete problem and the complexities of almost all the available algorithms are very high. On the other hand, the algorithms that have lesser complexity, does not provide an optimal solution. In this paper, we put forth a technique for coloring a graph with minimum number of colors and in significantly lesser time than any other technique by processing the edge table we use to represent a graph.

We introduce a concept of edge-distinguishing colourings of graphs. A closed neighbourhood of an edge $${e\in E(G)}$$ e ? E ( G ) is a subgraph N[e] induced by e and all edges adjacent to it. We say that a colouring c : E(G) ? C does not distinguish two edges e 1 and e 2 if there exists an isomorphism ? of N[e 1] onto N[e 2] such that ?(e 1) = e 2 and ? preserves colours of c. An edge-distinguishing index of a graph G is the minimum number of colours in a proper colouring which distinguishes every two distinct edges of G. We determine the edge-distinguishing index for cycles, paths and complete graphs.

Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colors needed to color the set of all directed paths in T, so that two paths of the same color never use the same directed arc of T, is equal to the maximum number of different paths that contain the same arc of T. The proof implies a polynomial time algorithm for actually coloring the paths with the minimum number of colors. When only a subset of the directed paths is to be colored, the problem is known to be NP-complete; we describe certain instances of the problem which can be efficiently solved. These results are applied to WDM (wavelength-division multiplexing) routing in all-optical networks. In particular, we solve the all-to-all gossiping problem in optical networks. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 183–196, 2001 An Extended Abstract of this work has been presented at ICALP'97 by L. Gargano [10].

A proper graph coloring is defined as coloring the nodes of a graph with the minimum number of colors without any two adjacent nodes having the same color. Dominator coloring of G is a proper coloring in which every vertex of G dominates every vertex of at least one color class. In this paper, new parameters, namely strong split and non-split dominator chromatic numbers and block, cycle, path non-split dominator chromatic numbers are introduced. These parameters are obtained for different classes of graphs and also interesting results are established.