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On Independent Domination in Planar Cubic Graphs

Abstract

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number, i(G), of G is the minimum cardinality of an independent dominating set. Goddard and Henning [Discrete Math. 313 (2013) 839–854] posed the conjecture that if G ∉ {K 3 , 3, C 5K 2} is a connected, cubic graph on n vertices, then i(G)38n, where C 5K 2 is the 5-prism. As an application of known result, we observe that this conjecture is true when G is 2-connected and planar, and we provide an infinite family of such graphs that achieve the bound. We conjecture that if G is a bipartite, planar, cubic graph of order n, then i(G)13n, and we provide an infinite family of such graphs that achieve this bound.

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On the Independence Number of Traceable 2-Connected Claw-Free Graphs

Abstract

A well-known theorem by Chvátal-Erdőos [A note on Hamilton circuits, Discrete Math. 2 (1972) 111–135] states that if the independence number of a graph G is at most its connectivity plus one, then G is traceable. In this article, we show that every 2-connected claw-free graph with independence number α(G) ≤ 6 is traceable or belongs to two exceptional families of well-defined graphs. As a corollary, we also show that every 2-connected claw-free graph with independence number α(G) ≤ 5 is traceable.

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Pancyclicity When Each Cycle Contains k Chords

Abstract

For integers nk ≥ 2, let c(n, k) be the minimum number of chords that must be added to a cycle of length n so that the resulting graph has the property that for every l ∈ {k, k + 1, . . . , n}, there is a cycle of length l that contains exactly k of the added chords. Affif Chaouche, Rutherford, and Whitty introduced the function c(n, k). They showed that for every integer k ≥ 2, c(n, k) ≥ Ωk(n1/ k) and they asked if n 1/ k gives the correct order of magnitude of c(n, k) for k ≥ 2. Our main theorem answers this question as we prove that for every integer k ≥ 2, and for sufficiently large n, c(n, k) ≤ kn 1/ k⌉ + k 2. This upper bound, together with the lower bound of Affif Chaouche et al., shows that the order of magnitude of c(n, k) is n 1/ k.

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The Path-Pairability Number of Product of Stars

Abstract

The study of a graph theory model of certain telecommunications network problems lead to the concept of path-pairability, a variation of weak linkedness of graphs. A graph G is k-path-pairable if for any set of 2k distinct vertices, si, ti, 1 ≤ ik, there exist pairwise edge-disjoint si, t i-paths in G, for 1 ≤ ik. The path-pairability number is the largest k such that G is k-path-pairable. Cliques, stars, the Cartesian product of two cliques (of order at least three) are ‘fully pairable’; that is ⌊n/2⌋-pairable, where n is the order of the graph. Here we determine the path-pairability number of the Cartesian product of two stars.

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Spectral Radius and Hamiltonicity of Graphs

Abstract

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or its complement, respectively. Secondly, we give the conditions for a nearly balanced bipartite graph to be traceable in terms of spectral radius, signless Laplacian spectral radius of the graph or its quasi-complement, respectively.

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Star Coloring Outerplanar Bipartite Graphs

Abstract

A proper coloring of the vertices of a graph is called a star coloring if at least three colors are used on every 4-vertex path. We show that all outerplanar bipartite graphs can be star colored using only five colors and construct the smallest known example that requires five colors.

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Total Roman Reinforcement in Graphs

Abstract

A total Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The minimum weight of a total Roman dominating function on a graph G is called the total Roman domination number of G. The total Roman reinforcement number rtR (G) of a graph G is the minimum number of edges that must be added to G in order to decrease the total Roman domination number. In this paper, we investigate the proper- ties of total Roman reinforcement number in graphs, and we present some sharp bounds for rtR (G). Moreover, we show that the decision problem for total Roman reinforcement is NP-hard for bipartite graphs.

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The Turań Number of 2P 7

Abstract

The Turán number of a graph H, denoted by ex(n, H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let Pk denote the path on k vertices and let mPk denote m disjoint copies of Pk. Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combin. Probab. Comput. 20 (2011) 837–853] determined the exact value of ex(n, kP) for large values of n. Yuan and Zhang [The Turán number of disjoint copies of paths, Discrete Math. 340 (2017) 132–139] completely determined the value of ex(n, kP 3) for all n, and also determined ex(n, Fm), where Fm is the disjoint union of m paths containing at most one odd path. They also determined the exact value of ex(n, P 3P 2 +1) for n ≥ 2 + 4. Recently, Bielak and Kieliszek [The Turán number of the graph 2P 5, Discuss. Math. Graph Theory 36 (2016) 683–694], Yuan and Zhang [Turán numbers for disjoint paths, arXiv:1611.00981v1] independently determined the exact value of ex(n, 2P 5). In this paper, we show that ex(n, 2P 7) = max{[n, 14, 7], 5n − 14} for all n ≥ 14, where [n, 14, 7] = (5n + 91 + r(r − 6))/2, n − 13 ≡ r (mod 6) and 0 ≤ r < 6.

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140 Years of Service to Science and Industry

Abstract

This compilation presents the main stages of the development process of the University of Óbuda over three centuries, from industrial education to higher education and finally, to the participants of the Conference. The first legal predecessor, the Secondary Industrial School (the Upper Industrial School located at the Vocational School), during the period of technics, led the way to the establishment of Donát Bánki and Kálmán Kandó, later to the Technical College of Light Industry, then to the establishment of an integrated Budapest Technical College, and onward to the successor, the University of Óbuda in the XXI. Century.

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Alternative Materials and Methodology in Contemporary Design

Abstract

Nowadays, concerns related to mankind’s increasing and destructive impact on the environment have influenced and changed the paradigms of product development; this in turn has brought about the appearance of environmental considerations in the creation and design of new products. Numerous industrial sectors have changed their processes of product development and production to meet the ecological requirements. Issues such as the scarcity of natural resources, increasing consumption and increasing pollution also present a number of problems. This article presents a process of comparing new alternatives with a specific methodology of decision-making. It is primarily focused on the use of rare natural materials and resources that are extracted and processed.

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