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A proper vertex coloring of a graph G is called a dynamic coloring if for every vertex ν with degree at least 2, the neighbors of ν receive at least two different colors. It was conjectured that if G is a regular graph, then χ2(G) - χ (G) ≤ 2. In this paper we prove that, apart from the cycles C4 and C5 and the complete bipartite graphs Kn,n, every strongly regular graph G, satisfies χ2(G) - χ (G) ≤ 1  

Let G be a graph. A zero-sum flow of G is an assignment of non-zero real numbers to the edges of G such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A zero-sum k-flow is a flow with values from the set {±1,...,±(k - 1)}. It has been conjectured that every r-regular graph, r ≥ 3, admits a zero-sum 5-flow. In this paper we provide an affirmative answer to this conjecture, except for r = 5  

Let R be a ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x, y∈R are adjacent if and only if x+y∈Z(R). Let the regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of total graph and regular graph of a commutative ring are contained in the set {3, 4, ∞}. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). Also, we prove that if R is a reduced left Noetherian ring and 2∈Z(R), then the chromatic number and the clique number of Reg(Γ(R)) are the... 

Let R be a ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by (TΓ (R)) is a graph with all elements of R as vertices, and two distinct vertices x, y in R are adjacent if and only if x + y Z(R). Let the regular graph of R, Reg (Γ(R)), be the induced subgraph of T(Γ (R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set { 3, 4,} . In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if R is a reduced left Noetherian ring and 2 Z(R), then the chromatic number and the clique number of Reg... 

Liu and Xu (1998) and Ellingham, Nam and Voss (2002) independently showed that every k-edge-connected simple graph G has a spanning tree T such that for each vertex v, dT(v) ≤ ⌈ d(v)/k ⌉ + 2. In this paper we show that every k-edge-connected graph G has a spanning tree T such that for each vertex v, dT(v)≤ ⌈ d(v)-2/k ⌉ + 2; also if G has k edge-disjoint spanning trees, then T can be found such that for each vertex v, dT(v) ≤ ⌈ d(v)-1/k ⌉ + 1. This result implies that every (r-1)-edge-connected r-regular graph (with r ≥ 4) has a spanning Eulerian subgraph whose degrees lie in the set {2,4,6}; also reduces the edge-connectivity needed for some theorems due to Barát and Gerbner (2014) and... 

For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet's Conjecture for bidirected graphs are equivalent. Among other results it is shown that if G is an r-regular graph (r ≥ 3), then G has a zero-sum 7-flow. Furthermore, if r is divisible by 3, then G has a zero-sum 5-flow. We also show a graph of order n with a zero-sum flow has a zero-sum (n + 3)2-flow. Finally, the existence of k-flows for small graphs is investigated  

A circular zero-sum flow for a graph G is a function f: E(G) → R { 0 } such that for every vertex v, ∑e∈Evf(e)=0, where Ev is the set of all edges incident with v. If for each edge e, 1 ≤ | f(e) | ≤ r- 1 , where r≥ 2 is a real number, then f is called a circular zero-sum r-flow. Also, if r is a positive integer and for each edge e, f(e) is an integer, then f is called a zero-sum r-flow. If G has a circular zero-sum flow, then the minimum r≥ 2 for which G has a circular zero-sum r-flow is called the circular zero-sum flow number of G and is denoted by Φ c(G). Also, the minimum integer r≥ 2 for which G has a zero-sum r-flow is called the flow number for G and is denoted by Φ (G). In this... 

A Deza graph G with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children GA and GB of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in GA or GB if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular. © 2021 Elsevier B.V  

Let F be a field, char (F) ≠ 2, and S ⊆ GLn (F), where n is a positive integer. In this paper we show that if for every distinct elements x, y ∈ S, x + y is singular, then S is finite. We conjecture that this result is true if one replaces field with a division ring. © 2009 Elsevier Inc. All rights reserved  

Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = RZ(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2n, where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices  

Let G be a graph. Assume that l and k are two natural numbers. An l-sum flow on a graph G is an assignment of non-zero real numbers to the edges of G such that for every vertex v of G the sum of values of all edges incident with v equals l. An l-sum k-flow is an l-sum flow with values from the set {±1,…, ±(k — 1)}. Recently, it was proved that for every r, r ≥ 3, r ≠ 5, every r-regular graph admits a 0-sum 5-flow. In this paper we settle a conjecture by showing that every 5-regular graph admits a 0-sum 5-flow. Moreover, we prove that every r-regular graph of even order admits a 1-sum 5-flow  

A graph G is called uniquelyk-list colorable (UkLC) if there exists a list of colors on its vertices, say L= { Sv∣ v∈ V(G) } , each of size k, such that there is a unique proper list coloring of G from this list of colors. A graph G is said to have propertyM(k) if it is not uniquely k-list colorable. Mahmoodian and Mahdian (Ars Comb 51:295–305, 1999) characterized all graphs with property M(2). For k≥ 3 property M(k) has been studied only for multipartite graphs. Here we find bounds on M(k) for graphs embedded on surfaces, and obtain new results on planar graphs. We begin a general study of bounds on M(k) for regular graphs, as well as for graphs with varying list sizes. © 2018, Springer... 

An induced path factor of a graph (Formula presented.) is a set of induced paths in (Formula presented.) with the property that every vertex of (Formula presented.) is in exactly one of the paths. The induced path number (Formula presented.) of (Formula presented.) is the minimum number of paths in an induced path factor of (Formula presented.). We show that if (Formula presented.) is a connected cubic graph on (Formula presented.) vertices, then (Formula presented.). Fix an integer (Formula presented.). For each (Formula presented.), define (Formula presented.) to be the maximum value of (Formula presented.) over all connected (Formula presented.) -regular graphs (Formula presented.) on... 

An induced path factor of a graph (Formula presented.) is a set of induced paths in (Formula presented.) with the property that every vertex of (Formula presented.) is in exactly one of the paths. The induced path number (Formula presented.) of (Formula presented.) is the minimum number of paths in an induced path factor of (Formula presented.). We show that if (Formula presented.) is a connected cubic graph on (Formula presented.) vertices, then (Formula presented.). Fix an integer (Formula presented.). For each (Formula presented.), define (Formula presented.) to be the maximum value of (Formula presented.) over all connected (Formula presented.) -regular graphs (Formula presented.) on... 

In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a Χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by d(G, Χ). We study the defining number of regular graphs. Let d(n,r, Χ -k) be the smallest defining number of all r-regular k-chromatic graphs with n vertices, and f(n, k) = k-2/2(k-1)n + 2+(k-2)(k-3)/2(k-1). Mahmoodian and Mendelsohn (1999) determined the value of d(n, k, Χ = k) for all k ≤ 5, except for the case of (n, k) = (10,5).... 

Let R be a ring and X R be a non-empty set. The regular graph of X, Γ(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Γ( GLn(F)) finite? In this paper, we show that if G is a soluble subgroup of GLn(F), then χ(Γ(G))<∞. Also, we show that for every field F, χ(Γ( Mn(F)))=χ(Γ( Mn(F(x)))), where x is an indeterminate. Finally, for every algebraically closed field F, we determine the maximum value of the clique number of Γ(), where denotes the subgroup generated by A∈ GLn(F)  

An induced path factor of a graph (Formula presented.) is a set of induced paths in (Formula presented.) with the property that every vertex of (Formula presented.) is in exactly one of the paths. The induced path number (Formula presented.) of (Formula presented.) is the minimum number of paths in an induced path factor of (Formula presented.). We show that if (Formula presented.) is a connected cubic graph on (Formula presented.) vertices, then (Formula presented.). Fix an integer (Formula presented.). For each (Formula presented.), define (Formula presented.) to be the maximum value of (Formula presented.) over all connected (Formula presented.) -regular graphs (Formula presented.) on... 

A graph is called equimatchable if all of its maximal matchings have the same size. Kawarabayashi, Plummer, and Saito showed that the only connected equimatchable 3-regular graphs are K4 and K3, 3. We extend this result by showing that for an odd positive integer r, if G is a connected equimatchable r-regular graph, then G ϵ {Kr+1, Kr,r}. Also it is proved that for an even r, a connected triangle-free equimatchable r-regular graph is isomorphic to one of the graphs C5, C7, and Kr,r. © 2017 Wiley Periodicals, Inc  

Let G be a 2k-edge-connected graph with 𝑘 ≥ 0 and let 𝐿(𝑣) ⊆ {𝑘,…, 𝑑𝐺(𝑣)} for every 𝑣 ∈ 𝑉 (𝐺). A spanning subgraph F of G is called an L-factor, if 𝑑𝐹 (𝑣) ∈ 𝐿(𝑣) for every 𝑣 ∈ 𝑉 (𝐺). In this article, we show that if (Formula presented.) for every 𝑣 ∈ 𝑉 (𝐺), then G has a k-edge-connected L-factor. We also show that if 𝑘 ≥ 1 and (Formula presented.) for every 𝑣 ∈ 𝑉 (𝐺), then G has a k-edge-connected L-factor. © 2018 Wiley Periodicals, Inc  

In this paper, we show that every (3k−3)-edge-connected graph G, under a certain degree condition, can be edge-decomposed into k factors G1,…,Gk such that for each vertex v∈V(Gi), |dGi(v)−dG(v)/k|<1, where 1≤i≤k. As an application, we deduce that every 6-edge-connected graph G can be edge-decomposed into three factors G1, G2, and G3 such that for each vertex v∈V(Gi) with 1≤i≤3, |dGi(v)−dG(v)/3|<1, unless G has exactly one vertex z with dG(z)⁄≡30. Next, we show that every odd-(3k−2)-edge-connected graph G can be edge-decomposed into k factors G1,…,Gk such that for each vertex v∈V(Gi), dGi(v) and dG(v) have the same parity and |dGi(v)−dG(v)/k|<2, where k is an odd positive integer and 1≤i≤k....