# On the Total {k}-Domination and Total {k}-Domatic Number of Graphs

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For a graph G = (V, E), an L(i, j) -labeling is a function f from the vertex set V to the set of all nonnegative integers such that f (x) - f ( y) â‰¥ i if d(x, y) = 1 and âˆ£f (x) - f (y)âˆ£ â‰¥ j if d(x, y) = 2, where d(x, y) denotes the distance between vertices x and y in G....

- Counting k-gons in finite projective planes. Voropaev, A. N. // Sibirskie Elektronnye Matematicheskie Izvestiia;2013, Vol. 10, p241
In the study of combinatorial properties of finite projective planes, an open problem is to determine whether the number of k-gons in a plane depends on its structure. For the values of k = 3; 4; 5; 6, the number of k-gons is a function of plane's order q only. By means of the explicit formulae...

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Let c( n) be the number of 4-colored partitions of n with two of the colors appearing only in multiples of 3. Chan and Cooper recently proved an infinite family of congruences for c( n). We prove more congruences by the theory of modular forms.

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We investigate the following weak Ramsey property of a cardinal Îº: If Ï‡ is coloring of nodes of the tree Îº by countably many colors, call a tree $${T \subseteq \kappa^{ < \omega}}$$ Ï‡-homogeneous if the number of colors on each level of T is finite. Write $${\kappa \rightsquigarrow...

- Upper Singed Domination Number of Graphs. Walikar, H. B.; Motammanavar, Satish V.; Venkatesh, T. // International Journal of Mathematical Combinatorics;Mar2014, Vol. 1, p87
A function f : V (G) â†’ {-1, 1} defined on the vertices of a graph G is a signed dominating function (SDF) if f(N[v]) â©¾ 1, âˆ€ v Ïµ V , where N[v] is the closed neighborhood of v. A SDF f is minimal if there does not exists signed dominating function g, g â‰ f such that...

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We define a biclique to be the complement of a bipartite graph, consisting of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to...

- Set Ideals with Complete Symmetry Group and Partition Ideals. Mishkin, Valery // Bulletin of the London Mathematical Society;1999, Vol. 31 Issue 6, p649
For a wide class of set ideals (including, for example, all uniform ideals), a criterion of completeness of their symmetry groups is provided in terms of ideal quotients (polars). We apply it to partition ideals, and derive the extended SierpiÅ„skiâ€“ErdÃ¶s duality principle. We...

- OPTIMAL ORIENTATIONS OF SUBGRAPHS OF COMPLETE BIPARTITE GRAPHS. LAKSHMI, R.; RAJASEKARAN, G.; SAMPATHKUMAR, R. // Transactions on Combinatorics;Mar2015, Vol. 4 Issue 1, p19
For a graph G, let D(G) be the set of all strong digraphs D obtained by the orientations of G: The orientation number of G is â†’ d(G) = min {d(D) âˆ£ D Ïµ D(G)}, where d(D) denotes the diameter of the digraph D: In this paper, we determine the orientation number for some subgraphs of...

- A Matrix Model with a Singular Weight and PainlevÃ© III. Brightmore, L.; Mezzadri, F.; Mo, M. // Communications in Mathematical Physics;Feb2015, Vol. 333 Issue 3, p1317
We investigate the matrix model with weight and unitary symmetry. In particular we study the double scaling limit as $${N \to \infty}$$ and $${(\sqrt{N} t, Nz^2 ) \to (u_1,u_2)}$$ , where N is the matrix dimension and the parameters ( u, u) remain finite. Using the Deift-Zhou steepest descent...