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Sylwia Cichacz, Bryan Freyberg and Dalibor Froncek

Abstract

Let G = (V, E) be a graph of order n. A distance magic labeling of G is a bijection : V → {1, 2, . . ., n} for which there exists a positive integer k such that ∑xN(v) (x) = k for all vV, where N(v) is the open neighborhood of v.

Tuttes flow conjectures are a major source of inspiration in graph theory. In this paper we ask when we can assign n distinct labels from the set {1, 2, . . ., n} to the vertices of a graph G of order n such that the sum of the labels on heads minus the sum of the labels on tails is constant modulo n for each vertex of G. Therefore we generalize the notion of distance magic labeling for oriented graphs.

Open access

Jafar Amjadi, Seyed Mahmoud Sheikholeslami and Marzieh Soroudi

Abstract

A total Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the following conditions: (i) every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 and (ii) the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function f is the value f(V (G)) = ΣuV(G)f (u). The total Roman domination number γtR(G) is the minimum weight of a total Roman dominating function of G. Ahangar et al. in [H.A. Ahangar, M.A. Henning, V. Samodivkin and I.G. Yero, Total Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2016) 501–517] recently showed that for any graph G without isolated vertices, 2γ(G) ≤ γtR(G) ≤ 3γ(G), where γ(G) is the domination number of G, and they raised the problem of characterizing the graphs G achieving these upper and lower bounds. In this paper, we provide a constructive characterization of these trees.

Open access

Zehui Shao, Seyed Mahmoud Sheikholeslami, Marzieh Soroudi, Lutz Volkmann and Xinmiao Liu

Abstract

Let G = (V, E) be a graph and let f : V (G) → {0, 1, 2} be a function. A vertex v is said to be protected with respect to f, if f(v) > 0 or f(v) = 0 and v is adjacent to a vertex of positive weight. The function f is a co-Roman dominating function if (i) every vertex in V is protected, and (ii) each vV with positive weight has a neighbor uV with f(u) = 0 such that the function fuv : V → {0, 1, 2}, defined by fuv(u) = 1, fuv(v) = f(v) − 1 and fuv(x) = f(x) for xV \ {v, u}, has no unprotected vertex. The weight of f is ω(f) = ∑vV f(v). The co-Roman domination number of a graph G, denoted by γcr(G), is the minimum weight of a co-Roman dominating function on G. In this paper, we give a characterization of graphs of order n for which co-Roman domination number is 2n3 or n − 2, which settles two open problem in [S. Arumugam, K. Ebadi and M. Manrique, Co-Roman domination in graphs, Proc. Indian Acad. Sci. Math. Sci. 125 (2015) 1–10]. Furthermore, we present some sharp bounds on the co-Roman domination number.

Open access

Mohammed Benatallah, Noureddine Ikhlef-Eschouf and Miloud Mihoubi

Abstract

A set of vertices S in a graph G = (V, E) is a dominating set if every vertex not in S is adjacent to at least one vertex in S. A domatic partition of graph G is a partition of its vertex-set V into dominating sets. A domatic partition 𝒫 of G is called b-domatic if no larger domatic partition of G can be obtained from 𝒫 by transferring some vertices of some classes of 𝒫 to form a new class. The minimum cardinality of a b-domatic partition of G is called the b-domatic number and is denoted by bd(G). In this paper, we explain some properties of b-domatic partitions, and we determine the b-domatic number of some families of graphs.

Open access

Norbert Polat

Abstract

We prove that any harmonic partial cube is antipodal, which was conjectured by Fukuda and K. Handa, Antipodal graphs and oriented matroids, Discrete Math. 111 (1993) 245–256. Then we prove that a partial cube G is antipodal if and only if the subgraphs induced by Wab and Wba are isomorphic for every edge ab of G. This gives a positive answer to a question of Klavžar and Kovše, On even and harmonic-even partial cubes, Ars Combin. 93 (2009) 77–86. Finally we prove that the distance-balanced partial cube that are antipodal are those whose pre-hull number is at most 1.

Open access

Yelena Mandelshtam

Abstract

In this paper we study graphs defined by pattern-avoiding words. Word-representable graphs have been studied extensively following their introduction in 2004 and are the subject of a book published by Kitaev and Lozin in 2015. Recently there has been interest in studying graphs represented by pattern-avoiding words. In particular, in 2016, Gao, Kitaev, and Zhang investigated 132-representable graphs, that is, word-representable graphs that can be represented by a word which avoids the pattern 132. They proved that all 132-representable graphs are circle graphs and provided examples and properties of 132-representable graphs. They posed several questions, some of which we answer in this paper.

One of our main results is that not all circle graphs are 132-representable, thus proving that 132-representable graphs are a proper subset of circle graphs, a question that was left open in the paper by Gao et al. We show that 123-representable graphs are also a proper subset of circle graphs, and are different from 132-representable graphs. We also study graphs represented by pattern-avoiding 2-uniform words, that is, words in which every letter appears exactly twice.

Open access

Lin Sun and Jianliang Wu

Abstract

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-planar graph with maximum degree ∆ ≥ 8 is edge-colorable with ∆ colors if each of its 5-cycles contains at most one chord.

Open access

Simon Špacapan

Abstract

The k-independence number of a graph G, denoted as αk(G), is the order of a largest induced k-colorable subgraph of G. In [S. Špacapan, The k-independence number of direct products of graphs, European J. Combin. 32 (2011) 1377–1383] the author conjectured that the direct product G × H of graphs G and H obeys the following bound

αk(G×H)αk(G)|V(H)|+αk(H)|V(G)|αk(G)αk(H),

and proved the conjecture for k = 1 and k = 2. If true for k = 3 the conjecture strenghtens the result of El-Zahar and Sauer who proved that any direct product of 4-chromatic graphs is 4-chromatic [M. El-Zahar and N. Sauer, The chromatic number of the product of two 4-chromatic graphs is 4, Combinatorica 5 (1985) 121–126]. In this paper we prove that the above bound is true for k = 3 provided that G and H are graphs that have complete tripartite subgraphs of orders α 3(G) and α 3(H), respectively.

Open access

Kefeng Diao, Fuliang Lu and Ping Zhao

Abstract

Let S be a finite set of positive integers. A mixed hypergraph ℋ is a onerealization of S if its feasible set is S and each entry of its chromatic spectrum is either 0 or 1. The minimum number of vertices, denoted by δ 3(S), in a 3-uniform bi-hypergraph which is a one-realization of S was determined in [P. Zhao, K. Diao and F. Lu, More result on the smallest one-realization of a given set, Graphs Combin. 32 (2016) 835–850]. In this paper, we consider the minimum number of edges in a 3-uniform bi-hypergraph which already has the minimum number of vertices with respect of being a minimum bihypergraph that is one-realization of S. A tight lower bound on the number of edges in a 3-uniform bi-hypergraph which is a one-realization of S with δ 3(S) vertices is given.

Open access

Lība Sokolovska, Alina Sultanova, Maksims Čistjakovs, Egils Cunskis and Modra Murovska

Abstract

The aim of this study was to investigate the possibility of using monocytes/macrophages as mediators in human herpesvirus-6 (HHV-6) infection of thyroid gland tissues in autoimmune thyroiditis (AIT). Seventy-three AIT patients were enrolled in this study. The control group consisted of 80 blood donors. Monocyte/macrophage isolation for AIT patient samples was performed by adherence. HHV-6 was detected in peripheral blood mononuclear cell (PBMC) DNA samples using nested polymerase chain reaction (nPCR). Gene expression of HHV-6 active infection marker (U79/80) and chemokine receptors (U12, U51) in patient monocyte/macrophage samples and blood donor PBMC samples was detected using reverse-transcription PCR. HHV-6 viral load was detected by using quantitative-PCR technique. The HHV-6 genomic sequence was found significantly more frequently among AIT patient than control group samples. Markers of active infection were found in 8 AIT patient monocyte/macrophage samples (11%) and in none of control group PBMC samples. HHV-6 U51 mRNA expression was detected only in AIT patient samples (2/24 previously positive for HHV-6). Since HHV-6 genomic sequences were found significantly more frequently in AIT patient samples and active infection markers were found in patient monocytes/macrophages, our results suggest that monocytes/macrophages may be used by HHV-6 as mediators for thyroid gland infection.