Edge Coloring

For positive integers k and d such that 4 ≤ k < d and k = 5, we determine the maximum number of rainbow colored copies of C 4 in a k-edge-coloring of the d-dimensional hypercube Q d . Interestingly, the k-edge-colorings of Q d yielding the maximum number of rainbow copies of C 4 also have the property that every copy of C 4 which is not rainbow is monochromatic.

We describe here a simple probabilistic model for graphs that are lifts of a fixed base graph G, i.e., those graphs from which there is a covering man onto G. Our aim is to investigate the properties of typical graphs in this class. In particular, we show that almost every lift of G is δ(G)-connected where δ(G) is the minimal

In this study, a new Hamiltonian loop construction algorithm is proposed for planar 3-regular diagrams of specific structures. Based on the original element theory and the perfect matching algorithm I proposed earlier, the algorithm establishes a deterministic Hamiltonian loop construction method by analyzing the local structural characteristics and global topological relationships of the graph. Theoretical analysis shows that for a specific structural plane 3-regular graph with a vertex number of 6k or 4i (k,i ≥ 1), the proposed algorithm can construct a Hamiltonian loop in linear time. The correctness of the algorithm is verified by rigorous mathematical proof, and its time complexity is significantly better than that of the Hamiltonian loop algorithm on the general graph. In addition, this study also reveals the intrinsic relationship between the Hamiltonian circuit, perfect matching, and graph coloring properties of this type of graph, which provides a new theoretical perspective for the study of the structural characteristics of specific graph classes. At the application level, the algorithm can be effectively applied to practical problems such as integrated circuit wiring optimization and network topology design. Experimental results show that compared with existing methods, the proposed algorithm has significant advantages in efficiency and scalability.

Indicated coloring is a type of game coloring in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by χ i (G). In this paper, we obtain structural characterization of connected {P 5 , K 4 , Kite, Bull}-free graphs which contains an induced C 5 and connected {P 6 , C 5 , K 1,3 }-free graphs that contains an induced C 6 . Also, we prove that {P 5 , K 4 , Kite, Bull}-free graphs that contains an induced C 5 and {P 6 , C 5 , P 5 , K 1,3 }free graphs which contains an induced C 6 are k-indicated colorable for all k ≥ χ(G). In addition, we show that K[C 5 ] is k-indicated colorable for all k ≥ χ(G) and as a consequence, we exhibit that {P 2 ∪ P 3 , C 4 }-free graphs, {P 5 , C 4 }-free graphs are k-indicated colorable for all k ≥ χ(G). This partially answers one of the questions which was raised by A. Grzesik in .

We consider edge colorings of graphs. An edge coloring is a majority coloring if for every vertex at most half of the edges incident with it are in one color. And edge coloring is a distinguishing coloring if for every non-trivial automorphism at least one edge changes its color. We consider these two notions together. We show that every graph without pendant edges has a majority distinguishing edge coloring with at most ⌈ √ ∆⌉ + 5 colors. Moreover, we show results for some classes of graphs and a general result for symmetric digraphs.

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The chromatic edge strength of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.

In a previous work, we proposed a new integer programming formulation for the graph coloring problem which, to a certain extent, avoids symmetry. We studied the facet structure of the 0/1-polytope associated with it. Based on these theoretical results, we present now a Branch-and-Cut algorithm for the graph coloring problem. Our computational experiences compare favorably with the well-known exact graph coloring algorithm DSATUR.

In this work we study the polytope associated with a 0,1-integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence that shows the efficacy of these inequalities used in a cutting-plane algorithm.

Precedence constraints are a part of a definition of any scheduling problem. After recalling, in precise graph-theoretical terms, the relations between task-on-arc and task-on-node representations, we show the equivalence of two distinct results for scheduling problems. Furthermore, again using these links between representations, we exhibit several new polynomial cases for various problems of scheduling preemptable tasks on unrelated parallel machines under arbitrary resource constraints.

A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order.

A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order.

A rectangular partition is a partition of a plane rectangle into an arbitrary number of nonoverlapping rectangles such that no four rectangles share a corner. In this note, it is proven that every rectangular partition admits a vertex coloring with four colors such that every rectangle, except possibly the outer rectangle, has all four colors on its boundary. This settles a conjecture of Dinitz et al. [Y. Dinitz, M.J. Katz, R. Krakovski, Guarding rectangular partitions, in: Abstracts 23rd Euro. Workshop Comput. Geom., 2007, pp. 30-33]. The proof is short, simple and based on 4-edge-colorability of a specific class of planar graphs.

Many NP-hard graph problems become easy for some classes of graphs, such as coloring is easy for bipartite graphs, but NP-hard in general. So we can ask question like when does a hard problem become easy? What is the minimum substructure for which the problem remains hard? We use the notion of boundary classes to study such questions. In this paper, we introduce a method for transforming the boundary class of one NP-hard graph problem into a boundary class for another problem. If Π and Γ are two NP-hard graph problems where Π is reducible to Γ , we transform a boundary class of Π into a boundary class of Γ. More formally if Π is reducible to Γ , where the reduction is bijective and it maps hereditary classes of graphs to hereditary classes of graphs, then X is a boundary class of Π if and only if the image of X under the reduction is a boundary class of Γ. This gives us a relationship between boundary classes and reducibility among several NP-hard problems. To show the strength of our main result, we apply our theorem to obtain some previously unknown boundary classes for a few graph problems namely; vertex-cover, clique, traveling-salesperson, bounded-degree-spanning-tree, subgraph-isomorphism and clique-cover.

Poset game, which includes some famous games. e.g., Nim and Chomp as sub-games, is an important two-player impartial combinatorial game. The rule of the game is as follows: For a given poset (partial ordered set), each player intern chooses an element and the selected element and it's descendants (elements succeeding it) are all removed from the poset. A player who choose the last element is the winner. On the complexity of poset game, although it is clearly in PSPACE, it have not known whether it is in P or NP-hard. Recently a weighted poset game, which is a generalization of poset game, have been presented and it was found that some sub-games of it can be solved in polynomial-time. The complexity of this game is also open. This paper shows that weighted poset game is PSPACE-complete even if the weights are restricted in {1, -1}, the dag, which represents the poset, is bipartite, and the length of each path in the dag is at most two.

A star edge coloring of a graph G is a proper edge coloring of G such that every path and cycle of length four in G uses at least three different colors. The star chromatic index of a graph G, is the smallest integer k for which G admits a star edge coloring with k colors. In this paper, we present a polynomial time algorithm that finds an optimum star edge coloring for every tree. We also provide some tight bounds on the star chromatic index of trees with diameter at most four, and using these bounds we find a formula for the star chromatic index of certain families of trees.

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that  every path and cycle of length four in $G$  uses at least three different colors. The star chromatic index of a graph $G$ is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors.  In this paper,  we present a polynomial time algorithm that finds an optimum star edge coloring for every tree. We also provide some tight bounds on the star chromatic index of trees with diameter at most four, and using these bounds we find a formula for the star chromatic index of certain families of trees.

The Berge-Fulkerson conjecture states that every bridgeless cubic graph can be covered with six perfect matchings such that each edge is covered exactly twice. An equivalent reformulation is that it's possible to find a 6-cycle 4-cover. In this paper we discuss the oriented version (o6c4c) of the latter statement, pose it as a conjecture and prove it for the family of Isaacs flower snarks. Similarly to the case of oriented cycle double cover, we can always construct an orientable surface (possibly with boundary) from an o6c4c solution. If the o6c4c solution itself splits into two (not necessarily oriented) cycle double covers, then it's also possible to build another pair of orientable surfaces (also possibly with boundaries). Finally we show how to build a ribbon graph, and for some special o6c4c cases we show that this ribbon graph corresponds to an oriented 6-cycle double cover. Github: https://github.com/gexahedron/cycle-double-covers

An edge-coloring of the complete graph K n we call F -caring if it leaves no F -subgraph of K n monochromatic and at the same time every subset of |V (F )| vertices contains in it at least one completely multicolored version of F . For the first two meaningful cases, when F = K 1,3 and F = P 4 we determine for infinitely many n the minimum number of colors needed for an F -caring edge-coloring of K n . An explicit family of 2⌈log 2 n⌉ 3-edgecolorings of K n so that every quadruple of its vertices contains a totally multicolored P 4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the five length cycle.

The dominated coloring of a graph G is a proper coloring of G such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of G is called the dominated chromatic number of G , denoted by χ dom (G). In this paper, dominated coloring of graphs is compared with (open) packing number of G and it is shown that if G is a graph of order n with diam(G) ≥ 3 , then χ dom (G) ≤ n − ρ(G) and if ρ0(G) = 2n/3 , then χ dom (G) = ρ0(G) , and if ρ(G) = n/2 , then χ dom (G) = ρ(G). The dominated chromatic numbers of the corona of two graphs are investigated and it is shown that if µ(G) is the Mycielsky graph of G , then we have χ dom (µ(G)) = χ dom (G) + 1. It is also proved that the Vizing-type conjecture holds for dominated colorings of the direct product of two graphs. Finally we obtain some Nordhaus-Gaddum-type results for the dominated chromatic number χ dom (G) .

We extend the definition of sandwich line-graphs, a class of auxiliary graphs the stable sets of which are in 1-to-1 correspondence with the colorings of the original graph, from graphs to partitioned graphs, this way, we obtain a one-to-one correspondence between stable sets and partition colorings.

We consider maximum properly edge-colored trees in edge-colored graphs G c. We also consider the problem where, given a vertex r, determine whether the graph has a spanning tree rooted at r, such that all root-to-leaf paths are properly colored. We consider these problems from graphtheoretic as well as algorithmic viewpoints. We prove their optimization versions to be NP-hard in general and provide algorithms for graphs without properly edge-colored cycles. We also derive some nonapproximability bounds. A study of the trends random graphs display with regard to the presence of properly edge-colored spanning trees is presented.

We introduce a number of problems regarding edge-color modifications in edge-colored graphs and digraphs. Consider a property π, a c-edge-colored graph G c not satisfying π, and an edge-recoloring cost matrix R = [rij]c×c where rij ≥ 0 denotes the cost of changing color i of edge e to color j. Basically, in this kind of problem the idea is to change the colors of one or more edges of G c in order to construct a new edge-colored graph such that the total edge-recoloring cost is minimized and property π is satisfied. We also consider the destruction of potentially undesirable structures with the minimum edge-recoloring cost. In this paper, we are especially concerned with the construction and destruction of properly edge-colored and monochromatic paths, trails and cycles in graphs and digraphs. Some related problems and future directions are presented.

This paper deals with the existence and search of properly edgecolored paths/trails between two, not necessarily distinct, vertices s and t in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s−t path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail * Sponsored by French Ministry of Education † Sponsored by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq 1 between s and t for some particular graphs and characterize edgecolored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k pairwise vertex/edge disjoint properly edge-colored s − t paths/trails in a c-edge colored graph G c is NP-complete even for k = 2 and c = O(n 2), where n denotes the number of vertices in G c. Moreover, we prove that these problems remain NP-complete for c-colored graphs containing no properly edgecolored cycles and c = O(n). We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.

Sufficient degree conditions for the existence of properly edge-colored cycles and paths in edge-colored graphs, multigraphs and random graphs are inverstigated. In particular, we prove that an edgecolored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to n+1 2 has properly edge-colored cycles of all possible lengths, including hamiltonian cycles. Longest properly edge-colored paths and hamiltonian paths between given vertices are considered as well.

In this paper we deal from an algorithmic perspective with different questions regarding monochromatic and properly edge-colored s-t paths/trails on edge-colored graphs. Given a c-edge-colored graph G c without properly edge-colored closed trails, we present a polynomial time procedure for the determination of properly edgecolored s-t trails visiting all vertices of G c a prescribed number of times. As an immediate consequence, we polynomially solve the Hamiltonian path (resp., Eulerian trail) problem for this particular class of graphs. In addition, we prove that to check whether G c contains 2 properly edge-colored s-t paths/trails with length at most L > 0 is NP-complete in the strong sense. Finally, we prove that, if G c is a general c-edge-colored graph, to find 2 monochromatic vertex disjoint s-t paths with different colors is NP-complete.

This paper deals with problems on non-oriented edge-colored graphs. The aim is to find a route between two given vertices s and t. This route can be a walk, a trail or a path. Each time a vertex is crossed by a walk there is an associated non-negative reload cost r i,j , where i and j denote, respectively, the colors of successive edges in this walk. The goal is to find a route whose total reload cost is minimum. Polynomial algorithms and proofs of NP-hardness are given for particular cases: when the triangle inequality is satisfied or not, when reload costs are symmetric (i.e., r i,j = r j,i) or asymmetric. We also investigate bounded degree graphs and planar graphs. We conclude the paper with the traveling salesman problem with reload costs.

In this article, I have tried to provide a comprehensive understanding of fundamental differences, historical evolution, and societal implications of analog and digital technologies. Analog technology, characterized by continuous signal representations of physical quantities, is contrasted with digital technology's binary nature. While digital technologies have surged in popularity, reshaping entire industries and daily life, analog technologies persist in niche applications. The historical narrative traces the digital revolution's inception from the introduction of the ENIAC computer in the 1940s to the miniaturization enabled by transistors in the 1950s. Mainframe computers, microprocessors, and the advent of personal computers in the 1970s and 1980s are pivotal milestones. The internet's emergence in the late 20th century and the proliferation of smartphones in the 21st century further demonstrate digital technology's transformative impact. I have also presented a case to show how digital and analog watches might have social and cultural implications, far beyond their technological nature.

This note is a report on a computer investigation of some small classical Ramsey numbers. We establish new lower bounds for the classical Ramsey numbers $R(3,11)$ and $R(4,8)$. In the first case, the bound is improved from $46$ (a record that had stood for 46 years) to $47$; and in the second case the bound is improved from $57$ to $58$.

The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author has found 37 new edge colorings of $K_{35}$ that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found using a heuristic search procedure.

We show that 28 s r(K4-e; 3) G 32. The construction used to establish the lower bound is made by using the strongly regular Schllfli graph for one of the colors, and then by partitioning its complement into two isomorphic graphs. The coloring has the property that each edge is on exactly one monochromatic K,, hence the vertex sets of these K,'s form a Steiner triple system.

Given a collection of matchings M = (M 1 , M 2 ,. .. , M q) (repetitions allowed), a matching M contained in M is said to be s-rainbow for M if it contains representatives from s matchings M i (where each edge is allowed to represent just one M i). Formally, this means that there is a function φ : M → [q] such that e ∈ M φ(e) for all e ∈ M , and |Im(φ)| s. Let f (r, s, t) be the maximal k for which there exists a set of k matchings of size t in some r-partite hypergraph, such that there is no s-rainbow matching of size t. We prove that f (r, s, t) 2 r−1 (s − 1), make the conjecture that equality holds for all values of r, s and t and prove the conjecture when r = 2 or s = t = 2. In the case r = 3, a stronger conjecture is that in a 3-partite 3-graph if all vertex degrees in one side (say V 1) are strictly larger than all vertex degrees in the other two sides, then there exists a matching of V 1. This conjecture is at the same time also a strengthening of a famous conjecture, described below, of Ryser, Brualdi and Stein. We prove a weaker version, in which the degrees in V 1 are at least twice as large as the degrees in the other sides. We also formulate a related conjecture on edge colorings of 3-partite 3-graphs and prove a similarly weakened version.

A semi-proper orientation of a given graph G, denoted by (D, w), is an orientation D with a weight function w : A(D) → Z + , such that the in-weight of any adjacent vertices are distinct, where the in-weight of v in D, denoted by w − D (v), is the sum of the weights of arcs towards v. The semi-proper orientation number of a graph G, denoted by − → χ s (G), is the minimum of maximum in-weight of v in D over all semi-proper orientation (D, w) of G. This parameter was first introduced by Dehghan (2019). When the weights of all edges eqaul to one, this parameter is equal to the proper orientation number of G. The optimal semi-proper orientation is a semi-proper orientation (D, w) such that max v∈V (G) w − D (v) = − → χ s (G). Araújo et al. (2016) showed that − → χ (G) ≤ 7 for every cactus G and the bound is tight. We prove that for every cactus G, − → χ s (G) ≤ 3 and the bound is tight. Araújo et al. (2015) asked whether there is a constant c such that − → χ (G) ≤ c for all outerplanar graphs G. While this problem remains open, we consider it in the weighted case. We prove that for every outerplanar graph G, − → χ s (G) ≤ 4 and the bound is tight.

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

A Grünbaum coloring of a triangulation is an assignment of colors to edges so that the edges about each face are assigned unique colors. In this paper we examine the color induced subgraphs given by a Grünbaum coloring of a triangulation and show that the existence ...

Let K,, be the complete graph with vertex set {u,, u2,. . . , u,,) and let g = (gl, . . . , g,,) be a sequence of positive integers. Color each edge of this K, red or blue. In this paper necessary and sufficient conditions are given which guarantee the existence of a connected span&g subgraph F in K,, (as colored) with both red degree and blue degree in F at vertex Ui eqa$ to gi. When each & = 1 this answers a question of Erdiis proved in this special case in [I].

We use the results of , to discuss the counting formulas of network flow polytopes and magic squares, i.e. the formula for the corresponding Ehrhart polynomial in terms of residues. We also discuss a description of the big cells using the theory of non broken circuit bases. The authors are partially supported by the Cofin 40 %, MIUR.

A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In 1985, Erdős and Nešetřil conjectured that every graph with maximum degree $\Delta$ has a strong edge-coloring using at most $\frac{5}{4}\Delta^2$ colors if $\Delta$ is even, and at most $\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd. Despite recent progress for large $\Delta$ by using an iterative probabilistic argument, the only nontrivial case of the conjecture that has been verified is when $\Delta = 3$, leaving the need for new approaches to verify the conjecture for any $\Delta\ge 4$. In this paper, we apply some ideas used in previous results to an upper bound of 21 for graphs with maximum degree 4, which improves a previous bound due to Cranston in 2006 and moves closer to the conjectured upper bound of 20.

We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree ∆ with max{∆, 9} colors. Thus the coloring is optimal for graphs with maximum degree ∆ ≥ 9. Moreover for ∆ = 4, 5, 6 we give linear-time algorithms that use ∆ + 2 colors. These results improve over the algorithms of Chrobak and Yung [1] and of Chrobak and Nishizeki [2] which color planar graphs using max{∆, 19} colors in linear time or using max{∆, 9} colors in O(n log n) time.

We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇r(G). For these classes we prove the existence of several partition results such as the existence of low tree-width and low treedepth colorings. This generalizes and simplifies several earlier results (obtained for minor closed classes).

We present a definition of the class NP in combinatorial context as the set of languages of structures defined by finitely many forbidden lifted substructures. We apply this to special syntactically defined subclasses and show how they correspond to naturally defined (and intensively studied) combinatorial problems. We show that some types of combinatorial problems like edge colorings and graph decompositions express the full computational power of the class NP. We then characterize Constraint Satisfaction Problems (i.e. H-coloring problems) which are expressible by finitely many forbidden lifted substructures. This greatly simplifies and generalizes the earlier attempts to characterize this problem. As a corollary of this approach we perhaps find a proper setting of Feder and Vardi analysis of CSP languages within the class MM-SNP.

Let G n,m be the grid [n]×[m]. G n,m is c-colorable if there is a function χ : G n,m → [c] such that there are no rectangles with all four corners the same color. We ask for which values of n, m, c is G n,m c-colorable? We determine (1) exactly which grids are 2-colorable, (2) exactly which grids are 3-colorable, (2) exactly which grids are 4-colorable. Our main tools are combinatorics and finite fields. Our problem has two motivations: (1) (ours) A Corollary of the Gallai-Witt theorem states that, for all c, there exists W = W (c) such that any c-coloring of [W ] × [W ] has a monochromatic square. The bounds on W (c) are enormous. Our relaxation of the problem to rectangles yields much smaller bounds. (2) Colorings grids to avoid a rectangle is equivalent to coloring the edges of a bipartite graph to avoid a monochromatic K 2,2,. Hence our work is related to bipartite Ramsey Numbers.

A star edge-coloring of a graph G is a proper edge coloring such that every 2-colored connected subgraph of G is a path of length at most 3. For a graph G, let the list star chromatic index of G, ch ′ st (G), be the minimum k such that for any k-uniform list assignment L for the set of edges, G has a star edge-coloring from L. Dvořák, Mohar and Šámal asked whether the list star chromatic index of every subcubic graph is at most 7. We prove that it is at most 8. We also prove that if the maximum average degree of a subcubic graph G is less than 7 3 respectively, 5 2 , then ch ′ st (G) ≤ 5 (respectively, ch ′ st (G) ≤ 6).

Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not.

The main aim of decision support systems is to find solutions that satisfy user requirements. Often, this leads to predictability of those solutions, in the sense that having the input data and the model, an adversary or enemy can predict to a great extent the solution produced by your decision support system. Such predictability can be undesirable, for example, in military or security timetabling, or applications that require anonymity. In this paper, we discuss the notion of solution predictability and introduce potential mechanisms to intentionally avoid it.

The anti-Ramsey number ar(G, H) with input graph G and pattern graph H, is the maximum positive integer k such that there exists an edge coloring of G using k colors, in which there are no rainbow subgraphs isomorphic to H in G. (H is rainbow if all its edges get distinct colors). The concept of anti-Ramsey number was introduced by Erdös, Simanovitz, and Sós in 1973. Thereafter several researchers investigated this concept in the combinatorial setting. The cases where pattern graph H is a complete graph Kr, a path Pr or a star K1,r for a fixed positive integer r, are well studied. Recently, Feng et al. revisited the anti-Ramsey problem for the pattern graph K1,t (for t ≥ 3) purely from an algorithmic point of view, due to its applications in interference modeling of wireless networks. They posed it as an optimization problem, the maximum edge q-coloring problem. For a graph G and an integer q ≥ 2, an edge q-coloring of G is an assignment of colors to edges of G, such that edges incident on a vertex span at most q distinct colors. The maximum edge q-coloring problem seeks to maximize the number of colors in an edge q-coloring of the graph G. Note that the optimum value of the edge q-coloring problem of G equals ar(G, K1,q+1). In this paper, we study ar(G, K1,t), the anti-Ramsey number of stars, for each fixed integer t ≥ 3, both from combinatorial and algorithmic point of view. The first of our main results presents an upper bound for ar(G, K1,q+1), in terms of number of vertices and the minimum degree of G. The second one improves this result for the case of triangle-free input graphs. For a positive integer t, let Ht denote a subgraph of G with maximum number of possible edges and maximum degree t. From an observation of Erdös, Simanovitz, and Sós, we get: |E(Hq−1)| + 1 ≤ ar(G, K1,q+1) ≤ |E(Hq)|. For instance, when q = 2, the subgraph E(Hq−1) refers to a maximum matching. It looks like |E(Hq−1)| is the most natural parameter associated with the anti-Ramsey number ar(G, K1,q+1), and the approximation algorithms for the maximum edge coloring problem usually proceed by first computing the Hq−1, then coloring all its edges with different colors and by giving one (sometimes more than one) extra colors to the remaining edges. The approximation guarantee of these algorithms usually depends on upper bounds for ar(G, K1,q+1) in terms of |E(Hq−1)|. Our third main result presents an upper bound for ar(G, K1,q+1) in terms of |E(Hq−1)|. All our results have algorithmic consequences. For some large special classes of graphs, such as d-regular graphs, where d ≥ 4, our results can be used to prove a better approximation guarantee for the well-known approximation algorithm for the maximum edge coloring problem. We also show that all our bounds are almost tight.

We study the minimum number of weights assigned to the edges of a graph G with no component K 2 so that any two adjacent vertices have distinct sets of weights on their incident edges. The best possible upper bound on this parameter is proved.

We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted χ [♯2] (G). This notion is related to other types of distancebased colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite planar graphs and subcubic K 4-minor-free graphs have exact square chromatic number at most 4. We then turn our attention to the class of fullerene graphs, which are cubic planar graphs with face sizes 5 and 6. We characterize fullerene graphs with exact square chromatic number 3. Furthermore, supporting a conjecture of Chen, Hahn, Raspaud and Wang (that all subcubic planar graphs are injectively 5-colorable) we prove that any induced subgraph of a fullerene graph has exact square chromatic number at most 5. This is done by first proving that a minimum counterexample has to be on at most 80 vertices and then computationally verifying the claim for all such graphs.

In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced matching. We consider intermediate types of edge-colorings, where some of the colors are allowed to form matchings, and the remaining form induced matchings. Our research is motivated by the conjecture proposed in a recent paper on S-packing edgecolorings (N. Gastineau and O. Togni, On S-packing edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019)). We prove that every graph with maximum degree 3 can be decomposed into one matching and at most 8 induced matchings, and two matchings and at most 5 induced matchings. We also show that if a graph is in class I, the number of induced matchings can be decreased by one, hence confirming the conjecture for this class of graphs.