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A proper vertex coloring of a graph G is called a dynamic coloring if for every vertex v of degree at least 2, the neighbors of v receive at least two different colors. Assume that ch2 (G) is the minimum number k such that for every list assignment of size k to each vertex of G, there is a dynamic coloring of G such that every vertex is colored with a color from its list. In this paper, it is proved that if G is a graph with no component isomorphic to C5 and Δ (G) ≥ 3, then ch2 (G) ≤ Δ (G) + 1, where Δ (G) is the maximum degree of G. This generalizes a result due to Lai, Montgomery and Poon which says that under the same assumptions χ2 (G) ≤ Δ (G) + 1. Among other results, we determine ch2... 

This study revisited the problem of online con ict-free coloring of intervals on a line, where each newly inserted interval must be assigned a color upon insertion such that the coloring remains conflict-free, i.e., for each point p in the union of the current intervals, there must be an interval I with a unique color among all intervals covering p. The bestknown algorithm uses O(log3 n) colors, where n is the number of current intervals. A simple greedy algorithm was presented that used only O(log n) colors. Therefore, the open problem raised in [Abam, M.A., Rezaei Seraji, M.J., and Shadravan, M. "Online conflictfree coloring of intervals", Journal of Scientia Iranica, 21(6), pp. 2138{2141... 

This study revisited the problem of online con ict-free coloring of intervals on a line, where each newly inserted interval must be assigned a color upon insertion such that the coloring remains conflict-free, i.e., for each point p in the union of the current intervals, there must be an interval I with a unique color among all intervals covering p. The bestknown algorithm uses O(log3 n) colors, where n is the number of current intervals. A simple greedy algorithm was presented that used only O(log n) colors. Therefore, the open problem raised in [Abam, M.A., Rezaei Seraji, M.J., and Shadravan, M. "Online conflictfree coloring of intervals", Journal of Scientia Iranica, 21(6), pp. 2138{2141... 

Let G and H be two graphs. A proper vertex coloring of G is called a dynamic coloring, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic coloring with k colors denoted by χ2(G). We denote the cartesian product of G and H by G□H. In this paper, we prove that if G and H are two graphs and δ(G) ≥ 2, then χ2(G□H) ≤ max(χ2(G),x(H)). We show that for every two natural numbers m and n, m,n ≥ 2, χ2(Pm□Pn) = 4. Also, among other results it is shown that if 3|mn, then χ2(C m□Cn) = 3 and otherwise χ2(C m□Cn) = 4  

In this paper, we study the problem of online conflict-free coloring of intervals on a line, where each newly inserted interval must be assigned a color upon insertion such that the coloring remains conflict-free, i.e. for each point p in the union of the current intervals, there must be an interval I with a unique color among all intervals covering p. We first present a simple algorithm which uses O(√n) colors where n is the number of current intervals. Next, we propose an CF-coloring of intervals which uses O(log3 n) colors  

Let G be a graph. The minimum number of colors needed to color the edges of G is called the chromatic index of G and is denoted by x0(G). It is well known that δ(G) ≤ x0(G) ≤ δ(G) + 1, for any graph G, whereδ(G) denotes the maximum degree of G. A graph G is said to be class 1 if x0(G) = δ(G) and class 2 if x0(G) = δ(G)+1. Also, Gδ is the induced subgraph on all vertices of degreeδ(G). Let f : V(G) ! N be a function. An f -coloring of a graph G is a coloring of the edges of E(G) such that each color appears at each vertex v 2 V(G) at most f (v) times. The minimum number of colors needed to f -color G is called the f -chromatic index of G and is denoted by x0f (G). It was shown that for every... 

Let G be a graph. The core of G, denoted by GΔ, is the subgraph of G induced by the vertices of degree Δ(G), where Δ(G) is the maximum degree of G. A k-edge coloring of a graph G is a function f:E(G)L, where |L|=k and f( e1)≠f( e2), for every two adjacent edges e1, e2 of G. The edge chromatic number of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class 1 if χ′(G)= Δ(G) and Class 2 if χ′(G)=Δ(G)+1. In this paper, it is shown that, for every connected graph of even order, if GΔ= C6, then G is Class 1. Also, we prove that, if G is a connected graph, and every connected component of GΔ is a unicyclic graph or a tree, and GΔ is not a... 

In this paper, we combine a novel Sequential Graph Coloring Heuristic Algorithm (SGCHA) with a non-systematic method based on a cultural algorithm to solve the graph coloring problem (GCP). The GCP involves finding the minimum number of colors for coloring the graph vertices such that adjacent vertices have distinct colors. In our solving approach, we first use an estimator which is implemented with SGCHA to predict the minimum colors. Then, in the non-systematic part which has been designed using cultural algorithms, we improve the prediction. Various components of the cultural algorithm have been implemented to solve the GCP with a self adaptive behavior in an efficient manner. As a result... 

Let $$G$$G be a graph. The core of $$G$$G, denoted by $$G_{Delta }$$GΔ, is the subgraph of $$G$$G induced by the vertices of degree $$Delta (G)$$Δ(G), where $$Delta (G)$$Δ(G) denotes the maximum degree of $$G$$G. A $$k$$k-edge coloring of $$G$$G is a function $$f:E(G)ightarrow L$$f:E(G)→L such that $$|L| = k$$|L|=k and $$f(e_1)e f(e_2)$$f(e1)≠f(e2), for any two adjacent edges $$e_1$$e1 and $$e_2$$e2 of $$G$$G. The chromatic index of $$G$$G, denoted by $$chi '(G)$$χ′(G), is the minimum number $$k$$k for which $$G$$G has a $$k$$k-edge coloring. A graph $$G$$G is said to be Class $$1$$1 if $$chi '(G) = Delta (G)$$χ′(G)=Δ(G) and Class $$2$$2 if $$chi '(G) = Delta (G) + 1$$χ′(G)=Δ(G)+1. Hilton... 

Suppose n colored points with k colors in Rd are given. The Smallest Color-Spanning Ball (SCSB) is the smallest ball containing at least one point of each color. As the computation of the SCSB in Lp metric (p≥1) is time-consuming, we focus on approximately computing the SCSB in near-linear time. Initially, we propose a 3-approximation algorithm running in O(n logn) time. This algorithm is then utilized to present a (1+ε)-approximation algorithm with the running time of O((1/ε)dn logn). We improve the running time to O((1/ε)dn) using randomized techniques. Afterward, spanning colors with two balls is studied. For a special case where d=1, there is an algorithm with O(n2) time. We demonstrate... 

Let G be a graph with no isolated vertex. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we provide a criterion under which a cubic graph has total domatic number at least two  

Let R be a commutative ring and Γ (R) be its zero-divisor graph. In this paper it is shown that for any finite commutative ring R, the edge chromatic number of Γ (R) is equal to the maximum degree of Γ (R), unless Γ (R) is a complete graph of odd order. In [D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, in: Lecture Notes in Pure and Appl. Math., Vol. 220, Marcel Dekker, New York, 2001, pp. 61-72] it has been proved that if R and S are finite reduced rings which are not fields, then Γ (R) ≃ Γ (S) if and only if R ≃ S. Here we generalize this result and prove that if R is a finite reduced ring which is not isomorphic to ℤ2 × ℤ 2 or to ℤ6 and S is a ring such that Γ (R) ≃ Γ (S), then R ≃... 

In 1967 Harary, Hedetniemi, and Prins showed that every graph G admits a complete t-coloring for every t with χ(G)≤t≤ψ(G), where χ(G) denotes the chromatic number of G and ψ(G) denotes the achromatic number of G which is the maximum number r for which G admits a complete r-coloring. Recently, Edwards and Rza̧żewski (2020) showed that this result fails for hypergraphs by proving that for every integer k with k≥9, there exists a k-uniform hypergraph H with a complete χ(H)-coloring and a complete ψ(H)-coloring, but no complete t-coloring for some t with χ(H)