Vertex-distinguishing Total Colorings of 2Cn

Let f be a proper total k-coloring of a simple graph G. For any vertex x ∈ V（G）, let Cf（x） denote the set of colors assigned to vertex x and the edges incident with x. If Cf（u） ≠ Cf（v） for all distinct vertices u and v of V（G）, then f is called a vertex- distinguishing total k-coloring of G. The minimum number k for which there exists a vertex- distinguishing total k-coloring of G is called the vertex-distinguishing total chromatic number of G and denoted by Xvt（G）. The vertex-disjoint union of two cycles of length n is denoted by 2Cn. We will obtain Xvt（2Cn） in this paper.

Adjacent vertex-distinguishing total colorings of K_s∨K_t

Let G be a simple graph and f be a proper total kcoloring of G. The color set of each vertex v of G is the set of colors appearing on v and the edges incident to v. The coloring f is said to be an adjacent vertex-distinguishing total coloring if the color sets of any two adjacent vertices are distinct. The minimum k for which such a coloring of G exists is called the adjacent vertex-distinguishing total chromatic number of G. The join graph of two vertex-disjoint graphs is the graph union of these two graphs together with all the edges that connect the vertices of one graph with the vertices of the other. The adjacent vertex-distinguishing total chromatic numbers of the join graphs of an empty graph of order s and a complete graph of order t are determined.

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K 7,n when7≤n≤95

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C（x） denote the set of colors of vertex x and of the edges incident with x, we call C（x） the color set of x. If C（u） ≠ C（v） for any two different vertices u and v of V （G）, then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.

Cyclically Interval Total Coloring of the One Point Union of Cycles

A total coloring of a graph G with colors 1, 2, ..., t is called a cyclically interval total t-coloring if all colors are used, and the edges incident to each vertex v∈V(G) together with v are colored by (dG(v)+1) consecutive colors modulo t, where dG(v) is the degree of the vertex v in G. The one point union ?of k-copies of cycle Cn is the graph obtained by taking v as a common vertex such that any two distinct cycles? and? are edge disjoint and do not have any vertex in common except v. In this paper, we study the cyclically interval total colorings of , where n≥3 and k≥2.

Cyclically Interval Total Colorings of Cycles and Middle Graphs of Cycles

A total coloring of a graph G is a functionsuch that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. A k-interval is a set of k consecutive integers. A cyclically interval total t-coloring of a graph G is a total coloring a of G with colors 1,2,...,t, such that at least one vertex or edge of G is colored by i,i=1,2,...,t, and for any, the set is a -interval, or is a -interval, where dG(x) is the degree of the vertex x in G. In this paper, we study the cyclically interval total colorings of cycles and middle graphs of cycles.

Vertex-distinguishing VE-total Colorings of Cycles and Complete Graphs

Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints.Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u.If C(u)=C(v) for any two vertices u and v of V (G),then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short.The minimum number of colors required for a VDVET coloring of G is denoted by χ ve vt (G) and it is called the VDVET chromatic number of G.In this paper we get cycle C n,path P n and complete graph K n of their VDVET chromatic numbers and propose a related conjecture.

Vertex-distinguishing IE-total Colorings of Cycles and Wheels

Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C（u）=C（v） for any two different vertices u and v of V （G）, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt（G）, and is called the VDIET chromatic number of G. We get the VDIET chromatic numbers of cycles and wheels, and propose related conjectures in this paper.

Adjacent Vertex-distinguishing E-total Coloring on Some Join Graphs Cm V Gn

Let G(V, E) be a simple connected graph and k be positive integers. A mapping f from V∪E to {1, 2, ··· , k} is called an adjacent vertex-distinguishing E-total coloring of G(abbreviated to k-AVDETC), if for uv ∈ E(G), we have f(u) ≠ f(v), f(u) ≠ f(uv), f(v) ≠ f(uv), C(u) ≠C(v), where C(u) = {f(u)}∪{f(uv)|uv ∈ E(G)}. The least number of k colors required for which G admits a k-coloring is called the adjacent vertex-distinguishing E-total chromatic number of G is denoted by x^e_(at) (G). In this paper, the adjacent vertexdistinguishing E-total colorings of some join graphs C_m∨G_n are obtained, where G_n is one of a star S_n , a fan F_n , a wheel W_n and a complete graph K_n . As a consequence, the adjacent vertex-distinguishing E-total chromatic numbers of C_m∨G_n are confirmed.

Vertex-distinguishing IE-total Colorings of Complete Bipartite Graphs K8,n

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.

Algorithm on the Optimal Vertex-Distinguishing Total Coloring of mC9

Let G be a simple graph and f be a proper total coloring(or a total coloring in brief) of G. For any vertex u in G, Cf(u) denote the set of colors of vertex u and edges which incident with vertex u. Cf(u) is said to be the color set of vertex u under f. If Cf(u) = Cf(v)for any two distinct vertices u and v of G, then f is called vertex-distinguishing total coloring of G(in brief VDTC), a vertex distinguishing total coloring using k colors is called k-vertexdistinguishing total coloring of G(in brief k-VDTC). The minimum number k for which there exists a k-vertex-distinguishing total coloring of G is called the vertex-distinguishing total chromatic number of G, denoted by χvt(G). By the method of prior distributing the color sets, we obtain vertex-distinguishing total chromatic number of m C9 in this paper.

Adjacent Vertex Distinguishing Total Coloring of M（Tn）

A k-proper total coloring of G is called adjacent distinguishing if for any two adjacent vertices have different color sets. According to the property of trees, the adjacent vertex distinguishing total chromatic number will be determined for the Mycielski graphs of trees using the method of induction.

On the (Δ + 2)-Total-Colorability of Planar Graphs with 7-Cycles Containing at Most Two Chords

The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords.

Adjacent Vertex Distinguishing I-total Coloring of Outerplanar Graphs

Let G be a simple graph with no isolated edge. An Ⅰ-total coloring of a graph G is a mapping φ : V(G) ∪ E(G) → {1, 2, · · ·, k} such that no adjacent vertices receive the same color and no adjacent edges receive the same color. An Ⅰ-total coloring of a graph G is said to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, we have C_φ(u) = C_φ(v), where C_φ(u) denotes the set of colors of u and its incident edges. The minimum number of colors required for an adjacent vertex distinguishing Ⅰ-total coloring of G is called the adjacent vertex distinguishing Ⅰ-total chromatic number, denoted by χ_at^i(G).In this paper, we characterize the adjacent vertex distinguishing Ⅰ-total chromatic number of outerplanar graphs.

Total phenolic, flavonoid content, and antioxidant activity of flour, noodles, and steamed bread made from different colored wheat grains by three milling methods

The objective of this study was to evaluate the effects of wheat variety, food processing, and milling method on antioxidant properties. Black wheat variety Heibaoshi 1 had the highest total phenolic content(659.8 μg gallic acid equivalents g-1), total flavonoid content(319.3 μg rutin equivalents g-1), and antioxidant activity, whereas light purple wheat variety Shandongzimai 1 had the lowest total flavonoid content(236.2 μg rutin equivalents g-1) and antioxidant activity. Whole wheat flour and partially debranned grain flour had significantly higher total phenolic contents, total flavonoid contents, and antioxidant activity than refined flour(P < 0.05). Compared with flour, total phenolic contents, total flavonoid contents and antioxidant activity decreased in noodles and steamed bread, whereas noodles had slightly higher total phenolic and flavonoid content than steamed bread. Antioxidant activities(by ferric reducing ability of plasma assay) of steamed bread made from whole wheat flour, partially debranned grain flour, and refined flour were 23.5%, 21.1%, and 31.6% lower, respectively, than the corresponding values of flour. These results suggested that black whole wheat flour and partially debranned grain flour are beneficial to human health.

Algorithm for reconstructing compressed sensing color imaging using the quaternion total variation

A new method for reconstructing the compressed sensing color image by solving an optimization problem based on total variation in the quaternion field is proposed, which can effectively improve the reconstructing ability of the color image. First, the color image is converted from RGB （red, green, blue） space to CMYK （cyan, magenta, yellow, black） space, which is assigned to a quaternion matrix. Meanwhile, the quaternion matrix is converted into the information of the phase and amplitude by the Euler form of the quatemion. Secondly, the phase and amplitude of the quatemion matrix are used as the smoothness constraints for the compressed sensing （CS） problem to make the reconstructing results more accurate. Finally, an iterative method based on gradient is used to solve the CS problem. Experimental results show that by considering the information of the phase and amplitude, the proposed method can achieve better performance than the existing method that treats the three components of the color image as independent parts.

Effects of Baking, Roasting and Frying on Total Polyphenols and Antioxidant Activity in Colored Chickpea Seeds

Chickpea lines with colored testa (seed coat) contain high levels of polyphenolic compounds that exhibit high levels of antioxidant activity. In a previous study, we showed that common processing procedures, such as soaking and cooking, decrease the levels of these bioactive compounds and subsequent overall antioxidant activity. The observed reduction in total phenolic content was due to the movement of polyphenols from the seed coat to the soaking or cooking water. Here, the effects of baking, roasting and frying processes were examined in relation to total phenolic content (TPC), total flavonoid content (TFC) and ferric-reducing ability of plasma antioxidant activity (FRAP AA) of colored chickpea seeds. Baked, fried and roasted colored chickpea seeds had significantly higher levels of TPC, TFC and FRAP AA than regular cream- and beige-colored seeds subjected to the same treatments. In contrast to our previous results with soaking and cooking, baking, frying and roasting retained most of the TPC, TFC and FRAP AA in the final products. Thus, colored chickpeas subjected to these three processing methods might be considered a functional food in addition to its traditional role of providing dietary proteins.

稀疏图的邻和可区别全列表染色

通过分析极小反例的结构,运用权转移方法,证明了最大度Δ(G)≥8且最大平均度小于3.2的图G的邻和可区别全选择数不超过Δ(G)+2.

Total Phenolic Content and Antioxidant Activity of Red and Yellow Quinoa (<i>Chenopodium quinoa</i>Willd.) Seeds as Affected by Baking and Cooking Conditions

Seeds with colored testa (seed coat) contain high concentrations of polyphenolic compounds that exhibit high levels of antioxidant activity. Common processing procedures, such as cooking and baking, decrease the levels of these bioactive compounds and consequently, overall antioxidant activity. Here, the effects of baking and cooking processes were examined on total phenolic content (TPC), total flavonoid content (TFC) and ferric-reducing ability of plasma antioxidant activity (FRAP AA) of red and yellow quinoa seeds. Our results indicate that red quinoa seed contains significantly higher levels of TPC, TFC and FRAP AA than yellow quinoa seeds. In addition, cooked and baked quinoa seeds retain most of their TPC, TFC and FRAP AA in the final product. Thus, red quinoa seeds processed by these two methods might be considered a functional food, in addition to its traditional role of providing dietary proteins. Due to their high antioxidant activity, red quinoa seeds might also contribute significantly to the management and/or prevention of degenerative diseases associated with free radical damage.

Total Phenolic Content and Antioxidant Activity of Chickpea (<i>Cicer</i><i>arietinum</i>L.) as Affected by Soaking and Cooking Conditions

Chickpea lines with colored testa (seed coat) contain high levels of polyphenolic compounds that exhibit high levels of antioxidant activity. However, common processing procedures, such as soaking and cooking, may decrease the levels of these bioactive compounds and subsequent overall antioxidant activity. Here, the effects of soaking, cooking and steaming processes were examined in relation to total phenolic content (TPC), total flavonoid content (TFC) and ferric reducing ability of plasma antioxidant activity (FRAP AA) of colored chickpea seeds. All processing steps significantly reduced TPC, TFC and FRAP AA in all of the tested chickpea seeds. Nevertheless, soaking the seeds at room temperature (for 22 h) resulted in a smaller decrease in TPC, TFC and FRAP AA than soaking at 60℃ (for 2 h). Moreover, steaming was superior to cooking in terms of conserving polyphenol and antioxidant activity. The observed reduction in TPC was mainly due to leaching of these compounds from the seed coat into the soaking or cooking water. Based on these results, we suggest that soaking at room temperature for 22 h followed by steaming for 1 h is the best method for retaining TPC, TFC and FRAP AA of colored chickpea.

Evaluation of revascularization after total arch replacement in common carotid artery occlusion

Occlusion of the common carotid artery(CCA) is rare. CCA occlusion(CCAO) can present as drowsiness and right hemiplegia related to emboli after total arch replacement. Although we selected a follow-up at first because color duplex sonography showed retrograde flow from the left external carotid artery to the internal carotid artery, this patient had epilepsy and single-photon emission computed tomography(SPECT) acquired quantitative results of actual brain perfusion and showed insufficient collateral blood flow. To improve brain perfusion, we performed a bypass of the left subclavian artery to left CCA bypass. Postoperatively, the patient did not have epilepsy and drowsiness. Also, right hemiplegia improved enough for him to walk with support. SPECT showed increased left cerebral flow(the asymmetry ratio was 71% to 81%). Evaluation of the carotid artery with color duplex sonography alone was insufficient when CCAO showed retrograde or collateral flow. We should have performed quantitative evaluation with SPECT at the same time.