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An edge-colored graph G is called properly connected if any two vertices are connected by a properly colored path. The proper connection number of a graph G , denoted by p c (G) , is the smallest number of colors that are needed to color G such that it is properly connected. A separating vertex of G is bad if it is contained in at least three blocks, each of which contains at most three vertices. Let G be a graph of order n without bad vertices. In this paper, we show that if the degree sums of any three nonadjacent vertices of a graph G is at least 3 n / 4 and n > 48 , then p c (G) = 2 , which generalizes the results due to Borozan et al. (2012) and Chang et al. (2019). [ABSTRACT FROM AUTHOR]

The generalized Turán number ex (n , K s , F) denotes the maximum number of copies of K s in an n -vertex F -free graph. Let kF denote k disjoint copies of F. Gerbner et al. (2019) [8] gave a lower bound for ex (n , K 3 , 2 C 5) and obtained the magnitude of ex (n , K s , k K r). In this paper, we determine the exact value of ex (n , K 3 , 2 C 5) and described the unique extremal graph for large n. Moreover, we also determine the exact value of ex (n , K r , (k + 1) K r) which generalizes some known results. [ABSTRACT FROM AUTHOR]

A Gallai coloring of a complete graph is a coloring of the edges without rainbow triangles (all edges colored differently). Given a graph H and a positive integer k , the Gallai Ramsey number G R k (H) is the minimum integer N such that every Gallai k -coloring of K N contains a monochromatic copy of H. A uniform coloring of a complete multipartite graph is a coloring of the edges such that the edges between any two parts receive the same color. Given a graph H and positive integers k and ℓ , the ℓ -uniform Ramsey number R k ℓ (H) is the minimum integer N such that any uniform k -coloring of any complete multipartite graph on N vertices with each part of cardinality no more than ℓ , contains a monochromatic copy of H. Let K m , n denote a complete bipartite graph. Wu et al. conjectured that G R k (K m , n) = (n − 1) (k − 2) + R 2 1 (K m , n) if R 2 1 (K m , n) ≥ 3 m − 2 , where k ≥ 3 and m ≥ n ≥ 2. In this paper, we show that G R k (K m , n) = (n − 1) (k − 2) + R 2 m − 1 (K m , n) for all integers k ≥ 3 and m ≥ n ≥ 2. Based on this result, we improve the known general bounds for G R k (K m , n) , and confirm the conjecture for some complete bipartite graphs. [ABSTRACT FROM AUTHOR]

Abstract For k given graphs G 1 , G 2 , ... , G k , k ≥ 2 , the k -color Ramsey number, denoted by R (G 1 , G 2 , ... , G k) , is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of G i colored with i , for some 1 ≤ i ≤ k. Let C m be a cycle of length m and K 1 , n a star of order n + 1. In this paper, firstly we give a general upper bound of R (C 4 , C 4 , ... , C 4 , K 1 , n). In particular, for the 3-color case, we have R (C 4 , C 4 , K 1 , n) ≤ n + 4 n + 5 + 3 and this bound is tight in some sense. Furthermore, we prove that R (C 4 , C 4 , K 1 , n) ≤ n + 4 n + 5 + 2 for all n = ℓ 2 − ℓ and ℓ ≥ 2 , and if ℓ is a prime power, then the equality holds. [ABSTRACT FROM AUTHOR]

A Hamiltonian cycle decomposition of a complete graph K 2 n + 1 is an n -edge-coloring of K 2 n + 1 such that each color class is a Hamiltonian cycle. For a given Hamiltonian cycle decomposition, a subgraph of K 2 n + 1 is called rainbow if all its edges are colored differently. Let H be any subgraph of K 2 n + 1 consisting of n edges. In 1999, Wu conjectured that K 2 n + 1 has a Hamiltonian cycle decomposition such that H is rainbow. In this paper, we show that the conjecture is true in the cases that | H | ≤ n + 1 , or H is a linear forest, or n ≤ 5. Using this result, we partially confirm a conjecture on restricted size Ramsey numbers due to Miralaei and Shahsiah in 2020. [ABSTRACT FROM AUTHOR]

For a graph G = ( V , E ) , the eccentric connectivity index of G , denoted by ξ c ( G ), is defined as ξ c ( G ) = ∑ v ∈ V ɛ ( v ) d ( v ) , where ɛ( v ) and d ( v ) are the eccentricity and the degree of v in G , respectively. In this paper, we first establish the sharp lower bound for the eccentric connectivity index in terms of the order and the minimum degree of a connected G , and characterize some extremal graphs, which generalize some known results. Secondly, we characterize the extremal trees having the maximum or minimum eccentric connectivity index for trees of order n with given degree sequence. Finally, we give a sharp lower bound for the eccentric connectivity index in terms of the order and the radius of a unicyclic G , and characterize all extremal graphs. [ABSTRACT FROM AUTHOR]

In this article, the kinetic stabilization processes of ultrathin polystyrene (PS) films with different molecular weights ( M w ) and interfacial conditions were monitored by controlling the cooling rate. We found that, for the high M w PS ultrathin film, the fictive temperature ( T f ) of liquid-cooled sample shows large change in a broad range of cooling rates while the kinetic stability proves no obvious variation. By using the low M w PS ultrathin film, the kinetic stability begins to show cooling rate dependence, which indicates the modification of kinetic stability of shorter polymer chains is easier. With the aid of fluorescence resonance energy transfer (FRET) method, the variations of inter-chain proximity of polymer chains with the cooling rate were investigated for both oligomer and high M w PS thin films, and the former is more remarkable. Given that the further the inter-chain proximity, the higher the degree of excess free volume, there are more space for the rearrangement of chains of low M w polymer ultrathin films to reach energy-minimum configurations during cooling. Additionally, the shift of T g,dyn after cooling can be enlarged or removed by introducing a mobile interface or burying the free surface. These results suggest that the kinetic stability of liquid-cooled ultrathin films may stay unchanged when fictive temperature varies obviously, and it can be modified by controlling the molecular weight of polymer and the interfacial condition. [ABSTRACT FROM AUTHOR]

Consider two graphs X and Y , each with n vertices. The friends-and-strangers graph FS (X , Y) of X and Y is a graph with vertex set consisting of all bijections σ : V (X) ↦ V (Y) , where two bijections σ , σ ′ are adjacent if and only if they differ precisely on two adjacent vertices of X , and the corresponding mappings are adjacent in Y. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Recently, Alon, Defant and Kravitz showed that if X and Y are two independent random graphs in G (n , p) , then the threshold probability guaranteeing the connectedness of FS (X , Y) is p 0 = n − 1 / 2 + o (1) , and suggested to investigate the general asymmetric situation, that is, X ∈ G (n , p 1) and Y ∈ G (n , p 2). In this paper, we show that if p 1 p 2 ≥ p 0 2 = n − 1 + o (1) and p 1 , p 2 ≥ w (n) p 0 , where w (n) → 0 as n → ∞ , then FS (X , Y) is connected with high probability, which extends the result on p 1 = p 2 = p , due to Alon, Defant and Kravitz. [ABSTRACT FROM AUTHOR]

For two given graphs G 1 and G 2 , the Ramsey number R ( G 1 , G 2 ) is the smallest integer N such that for any graph of order N , either G contains a copy of G 1 or its complement contains a copy of G 2 . Let C m be a cycle of length m and K 1 , n a star of order n + 1 . Parsons (1975) shows that R ( C 4 , K 1 , n ) ≤ n + ⌊ n − 1 ⌋ + 2 and if n is the square of a prime power, then the equality holds. In this paper, by discussing the properties of polarity graphs whose vertices are points in the projective planes over Galois fields, we prove that R ( C 4 , K 1 , q 2 − t ) = q 2 + q − ( t − 1 ) if q is an odd prime power, 1 ≤ t ≤ 2 ⌈ q 4 ⌉ and t ≠ 2 ⌈ q 4 ⌉ − 1 , which extends a result on R ( C 4 , K 1 , q 2 − t ) obtained by Parsons (1976). [ABSTRACT FROM AUTHOR]

The generalized Turán number ex (n , K s , H) is defined to be the maximum number of copies of a complete graph K s in any H -free graph on n vertices. Let F be a linear forest consisting of k paths of orders ℓ 1 , ℓ 2 ,... , ℓ k. By characterizing the structure of the F -free graph with large minimum degree, we determine the value of ex (n , K s , F) for n = Ω (| F | s) and k ≥ 2 except some ℓ i = 3 , and the corresponding extremal graphs. Our result confirms a conjecture, due to Yuan and Zhang (2021), on classical Turán number of any linear forest for n = Ω (| F | 2) and each ℓ i ≠ 3 , and improves some known results on classical Turán number of linear forest. [ABSTRACT FROM AUTHOR]

We consider proper edge colorings of a graph G using colors of the set { 1 , … , k } . Such a coloring is called neighbor sum distinguishing if for any u v ∈ E ( G ) , the sum of colors of the edges incident to u is different from the sum of the colors of the edges incident to v . The smallest value of k in such a coloring of G is denoted by ndi Σ ( G ) . Let mad ( G ) and Δ ( G ) denote the maximum average degree and the maximum degree of a graph G , respectively. In this paper we show that, for a graph G without isolated edges, if mad ( G ) < 8 3 , then ndi Σ ( G ) ≤ max { Δ ( G ) + 1 , 7 } ; and if mad ( G ) < 3 , then ndi Σ ( G ) ≤ max { Δ ( G ) + 2 , 7 } . [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n ≥ 3 . The remoteness ρ = ρ ( G ) is the maximum, over all vertices, of the average distance from a vertex to all others. The radius r = r ( G ) is the minimum, over all vertices, of the eccentricity of a vertex. Aouchiche and Hansen (2011) conjectured that ρ − r ≥ 3 − n 4 if n is odd and ρ − r ≥ 2 n − n 2 4 ( n − 1 ) if n is even. In this paper, we confirm this conjecture. In addition, we completely characterize extremal graphs attaining the lower bound. [ABSTRACT FROM AUTHOR]

Abstract: For two given graphs and , the planar Ramsey number is the smallest integer such that for any planar graph on vertices, either contains or its complement contains . Let denote a cycle of length and a complete graph of order . Sun, Yang, Lin and Song conjectured that and the conjecture was proved for . In this paper, it is shown that which confirms the conjecture for . [Copyright &y& Elsevier]

Isolated toughness of graph G is formulated by minimizing the ratio | S | / i (G − S) over all S ⊆ V (G) with i (G − S) ≥ 2 , where i (G − S) is the number of isolated vertices after removing vertex subset S from G. The previous works reveal that there exist explicit correlations between isolated toughness and fractional critical graphs (a.k.a. the graph admits a fractional factor after deleting given number of vertices). However, among the existing isolated toughness bounds, the term with respect to n (the number of removed vertices) is always at least n. In this paper, the exactly sharp isolated toughness bound for fractional (k , n) -critical graph is determined which reveals that the coefficient of term with regard to n can be reduced to 1/2. It is well acknowledged that some tight toughness related bounds can reach to the extreme value, while others cannot. We give an explanation why these tight bounds in various settings possess such pattern differences. [ABSTRACT FROM AUTHOR]

Abstract: A graph is 3-domination-critical (3-critical, for short), if its domination number is 3 and the addition of any edge decreases by 1. In this paper, we show that every 3-critical graph with independence number 4 and minimum degree 3 is Hamilton-connected. Combining the result with those in [Y.J. Chen, F. Tian, B. Wei, Hamilton-connectivity of 3-domination critical graphs with , Discrete Mathematics 271 (2003) 1–12; Y.J. Chen, F. Tian, Y.Q. Zhang, Hamilton-connectivity of 3-domination critical graphs with , European Journal of Combinatorics 23 (2002) 777–784; Y.J. Chen, T.C.E. Cheng, C.T. Ng, Hamilton-connectivity of 3-domination critical graphs with , Discrete Mathematics 308 (2008) (in press)], we solve the following conjecture: a connected 3-critical graph is Hamilton-connected if and only if , where is the toughness of . [Copyright &y& Elsevier]

Abstract: For two given graphs and , the Ramsey number is the smallest integer such that for any graph of order , either contains or the complement of contains . Let denote a cycle of length and a complete graph of order . In this paper, it is shown that . [Copyright &y& Elsevier]

For a graph G and a family of graphs H , the Turán number e x (G , H) is defined to be the maximum number of edges among all H -free subgraphs of G. Inverting this problem, Briggs and Cox (2019) [5] studied the extremal function ε H (k) = s u p { e (G) | e x (G , H) < k } , where e (G) is the size of G , and suggested to investigate the extremal function φ H (k) = s u p { χ (G) | e x (G , H) < k } , where χ (G) denotes the chromatic number of G. Let K n be a complete graph of order n and H a given graph. In this paper, we establish a tight general upper bound for φ H (k) and conjecture φ H (k) = m a x { n | e x (K n , H) < k } for H ≠ 2 K 2. We also confirm this conjecture for many instances of H. [ABSTRACT FROM AUTHOR]

Abstract: Let be a graph. A signed dominating function on G is a function such that for each , where is the closed neighborhood of . The weight of a signed dominating function f is . A signed dominating function f is minimal if there exists no signed dominating function g such that and for each . The upper signed domination number of a graph G, denoted by , equals the maximum weight of a minimal signed dominating function of G. In this paper, we establish an tight upper bound for in terms of minimum degree and maximum degree. Our result is a generalization of those for regular graphs and nearly regular graphs obtained in [O. Favaron, Signed domination in regular graphs, Discrete Math. 158 (1996) 287–293] and [C.X. Wang, J.Z. Mao, Some more remarks on domination in cubic graphs, Discrete Math. 237 (2001) 193–197], respectively. [Copyright &y& Elsevier]

Abstract: A graph G is 3-domination critical if its domination number is 3 and the addition of any edge decreases by 1. Let G be a 3-domination critical graph with toughness more than one. It was proved that G is Hamilton-connected for the cases [Y.J. Chen, F. Tian, B. Wei, Hamilton-connectivity of 3-domination critical graphs with , Discrete Math. 271 (2003) 1–12] and [Y.J. Chen, F. Tian, Y.Q. Zhang, Hamilton-connectivity of 3-domination critical graphs with , European J. Combin. 23 (2002) 777–784]. In this paper, we show G is Hamilton-connected for the case . [Copyright &y& Elsevier]

Abstract: For two given graphs and , the Ramsey number is the smallest integer such that for any graph of order , either contains or the complement of contains . Let denote a cycle of length m and a complete graph of order . It was conjectured that for and . We show that and , and the latter result confirms the conjecture in the case when . [Copyright &y& Elsevier]

Abstract: A subset of the vertex set of a graph is called acyclic if the subgraph it induces in contains no cycles. is called an acyclic dominating set of if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by , is called the acyclic domination number of . Hedetniemi et al. [Acyclic domination, Discrete Math. 222 (2000) 151–165] introduced the concept of acyclic domination and posed the following open problem: if is the minimum degree of , is for any graph whose diameter is two? In this paper, we provide a negative answer to this question by showing that for any positive , there is a graph with diameter two such that . [Copyright &y& Elsevier]

Abstract: A graph G is said to be cyclable if for each orientation D of G, there exists a set such that reversing all the arcs with one end in S results in a hamiltonian digraph. Let G be 4-connected simple graph of even order . In this paper, we show that if for any with , then G is cyclable. [Copyright &y& Elsevier]

Abstract: Let be a graph. A subset of is called a dominating set if each vertex of has at least one neighbor in . The domination number equals the minimum cardinality of a dominating set in . A minus dominating function on is a function such that for each , where is the closed neighborhood of . The minus domination number of is . It was incorrectly shown in [X. Yang, Q. Hou, X. Huang, H. Xuan, The difference between the domination number and minus domination number of a cubic graph, Applied Mathematics Letters 16 (2003) 1089–1093] that there is an infinite family of cubic graphs in which the difference can be made arbitrary large. This note corrects the mistakes in the proof and poses a new problem on the upper bound for in cubic graphs. [Copyright &y& Elsevier]

Let Tn denote a tree of order n and Wm a wheel of order m + 1. In a previous paper, we evaluated the Ramsey number R(Tn, Wm) in the cases where Tn is the star of order n and m = 6 or m is odd and n ≥ m - 1 ≥ 2. In this paper, we determine R(Tn, W6) in the case where the maximum degree of Tn is at least n - 3. Our results show that a recent conjecture of Baskoro et al. is false. [Copyright &y& Elsevier]

A graph G is 3-domination critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. Let G be a 3-connected 3-domination critical graph with α(G)&les;δ(G). In this paper, we show that G is Hamilton-connected if and only if τ(G)>1, where τ(G) is the toughness of G. [Copyright &y& Elsevier]

A graph G is 3-domination-critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. Wojcicka conjectured that every 3-domination-critical graph with δ&ges;2 has a hamiltonian cycle (J. Graph Theory 14 (1990) 205–215). The conjecture had been proved and its proof consists of two parts: the case α&les;δ+1 (J. Graph Theory 25 (1997) 173–184) and the case α=δ+2 (Discrete Appl. Math. 92 (1999) 57–70). In this paper, we give a new and simple proof of the conjecture by using Hanson's (J. Combin. Math. Combin. Comput. 13 (1993) 121–128) and Bondy-Chva´tal's (Discrete Math. 15(1976) 111–135) closure operations. [Copyright &y& Elsevier]

A cycle C of a graph G is dominating if any vertex of V (G) ∖ V (C) has at least one neighbor on C and V (G) ∖ V (C) is an independent set. Let G be a k -connected graph of order n ≥ 3 with k ≥ 2. In this paper, we show that every longest cycle of G is dominating if the degree sums is more than (k + 1) (n + 1) ∕ 3 for any k + 1 pairwise nonadjacent vertices, and the lower bound is sharp, which generalizes the results due to Bondy (1980) for k = 2 and Lu et al. (2005) for k = 3. [ABSTRACT FROM AUTHOR]

Abstract: For two given graphs and , the Ramsey number is the smallest integer such that for any graph of order , either contains or the complement of contains . Let denote a cycle of order and a wheel of order . In this paper, we show that for odd, and , which was conjectured by Surahmat, Baskoro and Tomescu. [Copyright &y& Elsevier]

Abstract: For two given graphs and , the Ramsey number is the smallest integer n such that for any graph G of order n, either G contains or the complement of G contains . Let denote a path of order n and a wheel of order . In this paper, we show that for m even and and for m odd and . [Copyright &y& Elsevier]

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k -coloring is a Gallai coloring that uses at most k colors. Given an integer k ≥ 2 , and graphs H 1 , H 2 , ... , H k , the Gallai-Ramsey number G R (H 1 , H 2 , ... , H k) is the least positive integer N such that every Gallai k -coloring of the complete graph K N contains a monochromatic copy of H i in color i for some i ∈ { 1 , 2 , ... , k }. Let m G denote the union of m disjoint copies of G. This paper studies the Gallai–Ramsey number G R (t 1 K 3 , ... , t k K 3) for all integers t 1 ≥ t 2 ≥ ⋯ ≥ t k ≥ 1. The case t 1 = t 2 = ⋯ = t k = 1 was obtained by Chung and Graham (1983). In this paper, we obtain a general bound of G R (t 1 K 3 , ... , t k K 3) for all t 1 ≥ t 2 ≥ ⋯ ≥ t k ≥ 1 , and prove that both the upper and lower bounds can be attained for some t i , i ∈ { 1 , ... , k }. [ABSTRACT FROM AUTHOR]

Abstract: For two given graphs and , the Ramsey number is the smallest integer such that for any graph of order , either contains or the complement of contains . Let denote a cycle of order and a wheel of order . It is conjectured by Surahmat, E.T. Baskoro and I. Tomescu that for even , and . In this paper, we confirm the conjecture for . [Copyright &y& Elsevier]

Let X and Y be any two graphs of order n. The friends-and-strangers graph FS (X , Y) of X and Y is a graph with vertex set consisting of all bijections σ : V (X) ↦ V (Y) , in which two bijections σ , σ ′ are adjacent if and only if they differ precisely on two adjacent vertices of X , and the corresponding images are adjacent in Y. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Let K k , n − k be a complete bipartite graph of order n. In 1974, Wilson characterized the connectedness of FS (K 1 , n − 1 , Y) by using algebraic methods. In this paper, by using combinatorial methods, we investigate the connectedness of FS (K k , n − k , Y) for any Y and all k ≥ 2 , including Y being a random graph, as suggested by Defant and Kravitz, and pose some open problems. [ABSTRACT FROM AUTHOR]

Recently, many exciting achievements have been made in improving the cycling performance of SnO 2 -based electrode. However, the low initial Coulombic efficiency of SnO 2 -based electrode caused by the formation of solid electrolyte interphase and some of the irreversible formation of Li 2 O in SnO 2 structure still remains unresolved. Here this work reports a primary-cell process to prelithiate the SnO 2 -based composites to compensate the irreversible loss during the first cycle. The primary-cell-prelithiated SnO 2 @C composites performs a high initial Coulombic efficiency from 92.7% to >100% with the prelithiation time extends. Even after 600 cycles, the prelithiated (24 h) SnO 2 @C composites show a high capacity retention of 95% (848 mAh g −1 ) at a current density of 0.5 A g −1 . [ABSTRACT FROM AUTHOR]

For bipartite graphs H 1 , ... , H μ , μ ≥ 2 , the μ -color bipartite Ramsey number, denoted by R b (H 1 , ... , H μ) , is the least positive integer N such that if we arbitrarily color the edges of a complete bipartite graph K N , N with μ colors, then it contains a monochromatic copy of H i in color i for some i , 1 ≤ i ≤ μ. Let C 4 and K 1 , n be a quadrilateral and a star on n + 1 vertices, respectively. In this paper, we show that the (μ + 1) -color bipartite Ramsey number R b (C 4 , ... , C 4 , K 1 , n) ≤ n + ⌈ 1 2 μ 2 (4 n + μ 2 + 2 μ − 7) + 4 ⌉ + μ 2 + μ 2 − 1. Moreover, using algebraic methods, we construct Ramsey graphs or near Ramsey graphs and determine infinitely many values of R b (C 4 , ... , C 4 , K 1 , n) , which reach the upper bound if μ = 1 , 2 and are at most ⌊ μ 2 ⌋ less than the upper bound if μ ≥ 3. [ABSTRACT FROM AUTHOR]

For graphs G 1 and G 2 , the Ramsey number R ( G 1 , G 2 ) is the smallest integer N such that, for any graph G of order N , G contains G 1 as a subgraph or the complement of G contains G 2 as a subgraph. Let T n denote a tree of order n , W n a wheel of order n + 1 and F n a fan of order 2 n + 1 . We establish Ramsey numbers for fans and trees versus wheels of even order, thereby extending several known results. In particular, we prove that R ( F n , W m ) = 6 n + 1 for odd m ≥ 3 and n ≥ ( 5 m + 3 ) / 4 , and that R ( T n , W m ) = 3 n − 2 for odd m ≥ 3 and n ≥ m − 2 , and T n being a tree for which the Erdős–Sós Conjecture holds. [ABSTRACT FROM AUTHOR]

A broom is a tree obtained by identifying an endpoint of a path with the center of a star. Let G be a connected graph of order n ≥ 3. Chen et al. [2] conjectured that if the degree sum is at least n − 2 for any three pairwise nonadjacent vertices, then G contains a spanning broom. In this paper, we confirm the conjecture for n > 50. [ABSTRACT FROM AUTHOR]

In this paper, we study the upper size Ramsey number u ( G 1 , G 2 ) , defined by Erdős and Faudree, as well as the star-critical Ramsey number r ∗ ( G 1 , G 2 ) , defined by Hook and Isaak. We define Ramsey-full graphs and size Ramsey good graphs, and perform a detailed study on these graphs. We generalize earlier results by determining u ( n K k , m K l ) and r ∗ ( n K k , m K l ) for k , l ≥ 3 and large m , n ; u ( C n , C m ) for m odd, with n > m ≥ 3 ; and r ∗ ( C n , C m ) for m odd, with n ≥ m ≥ 3 and ( m , n ) ≠ ( 3 , 3 ) . [ABSTRACT FROM AUTHOR]

For two given graphs G 1 and G 2 , the Ramsey number R ( G 1 , G 2 ) is the smallest integer N such that, for any graph G of order N , either G contains G 1 as a subgraph or the complement of G contains G 2 as a subgraph. Let T n be a tree of order n , S n a star of order n , and F m a fan of order 2 m + 1 , i.e., m triangles sharing exactly one vertex. In this paper, we prove that R ( T n , F m ) = 2 n − 1 for n ≥ 3 m 2 − 2 m − 1 , and if T n = S n , then the range can be replaced by n ≥ max { m ( m − 1 ) + 1 , 6 ( m − 1 ) } , which is tight in some sense. [ABSTRACT FROM AUTHOR]

For a graph H and an integer k ≥ 1 , the k -color Ramsey number R k (H) is the least integer N such that every k -coloring of the edges of the complete graph K N contains a monochromatic copy of H. Let C m denote the cycle on m ≥ 4 vertices and let Θ m denote the family of graphs obtained from C m by adding an additional edge joining two non-consecutive vertices. Unlike Ramsey number of odd cycles, little is known about the general behavior of R k (C 2 n) except that R k (C 2 n) ≥ (n − 1) k + n + k − 1 for all k ≥ 2 and n ≥ 2. In this paper, we study Ramsey number of even cycles with chords under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. For an integer k ≥ 1 , the Gallai-Ramsey number G R k (H) of a graph H is the least positive integer N such that every Gallai k -coloring of the complete graph K N contains a monochromatic copy of H. We prove that G R k (Θ 2 n) = (n − 1) k + n + 1 for all k ≥ 2 and n ≥ 3. This implies that G R k (C 2 n) = (n − 1) k + n + 1 all k ≥ 2 and n ≥ 3. Our result yields a unified proof for the Gallai-Ramsey number of all even cycles on at least four vertices. [ABSTRACT FROM AUTHOR]

Abstract: Given two graphs and , the Ramsey number is the smallest integer such that for any graph of order , either contains or its complement contains . Let denote a cycle of order and a wheel of order . In this paper, it is shown that for odd, and , and for odd and . [Copyright &y& Elsevier]

To meet the requirements for the quick charge for the hybrid electric vehicles (HEVs) or plug-in hybrid electric vehicles (PHEVs), better rate capability is urgently needed for the lithium ion batteries (LIBs). Here in our work, a new anode with excellent rate capability is developed. In this anode, nanoporous iron oxide nanoparticles coated with carbon (designated as Fe 2 O 3 @C NPs) are homogeneous distributed in carbon nanofibers (CNFs), which can be designated as Fe 2 O 3 @C CNFs. The CNF constrains the nanoporous Fe 2 O 3 @C NPs along the longitudinal direction and the fibers are cross-linked, establishing a three dimensional (3D) stable electric conductive network. The nanoporous Fe 2 O 3 @C NPs exposed on the surface of CNFs provide more active sites for electrode reactions. The thin carbon shell around the Fe 2 O 3 NPs gives the additional protection and improves the conductive connection between the nanofibers, leading to an integral conductive network which links all the Fe 2 O 3 @C NPs. The Fe 2 O 3 @C CNFs structure exhibits good capacity retention and excellent rate capacity at high current density. [ABSTRACT FROM AUTHOR]