Universally signable graphs

Discussiones Mathematicae Graph Theory, 2015

The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede's relatively difficult characterization of consistent vertex-signed graphs. We give a simple proof that does not depend on Hoede's theorem as well as a structural description of line-consistent signed graphs.

arXiv: Combinatorics, 2013

An edge uv in a graph \Gamma\ is directionally 2-signed (or, (2,d)-signed) by an ordered pair (a,b), a,b in {+,-}, if the label l(uv) = (a,b) from u to v, and l(vu) = (b,a) from v to u. Directionally 2-signed graphs are equivalent to bidirected graphs, where each end of an edge has a sign. A bidirected graph implies a signed graph, where each edge has a sign. We extend a theorem of Sriraj and Sampathkumar by proving that the signed graph is antibalanced (all even cycles and only even cycles have positive edge sign product) if, and only if, in the bidirected graph, after suitable reorientation of edges every vertex is a source or a sink.

Cubo (Temuco), 2020

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, • • • , 2q − 1} such that the induced function f * : E(G) → {1, 3, • • • , 2q − 1} defined by f * (uv) = f (u) + f (v) is a bijection. In this paper we prove that T p-tree, T•P m , T•2P m , regular bamboo tree, C n• P m , C n• 2P m and subdivided grid graphs are odd harmonious. RESUMEN Un grafo G(p, q) se dice impar armonioso si existe una inyección f : V (G) → {0, 1, 2, • • • , 2q − 1} tal que la función inducida f * : E(G) → {1, 3, • • • , 2q − 1} definida por f * (uv) = f (u) + f (v) es una biyección. En este artículo probamos que los grafos T p-árboles, T•P m , T•2P m ,árboles bambú regulares, C n• P m , C n• 2P m y cuadrículas subdivididas son impar armoniosos.

Mapana - Journal of Sciences, 2012

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V, E), called the underlying graph of S and σ : E → {+1, −1} is a function from the edge set E of Su into the set {+1, −1}, called the signature of S. A sigraph S is sign-compatible if there exists a marking µ of its vertices such that the end vertices of every negative edge receive ‘−1’ marks in µ and no positive edge does so. In this paper, we characterize S such that its ×-line sigraphs, semi-total line sigraphs, semi-total point sigraphs and total sigraphs are sign-compatible.

Journal of Graph Theory, 2018

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A banner is a graph which consists of a hole on four vertices and a single vertex with precisely one neighbor on the hole. We prove that a (banner, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or contains a homogeneous set. Using this structure result, we design a polynomial-time algorithm for recognizing (banner, odd hole)-free graphs. We also design polynomialtime algorithms to find, for such a graph, a minimum coloring and largest stable set. A graph G is perfectly divisible if every induced subgraph H of G contains a set X of vertices such that X meets all largest cliques of H, and X induces a perfect graph. The chromatic number of a perfectly divisible graph G is bounded by ω 2 where ω denotes the number of vertices in a largest clique of G. We prove that (banner, odd hole)-free graphs are perfectly divisible.

Graphs and Combinatorics, 2018

A hole is an induced cycle with at least four vertices. A hole is even if it has an even number of vertices. Even-hole-free graphs are being actively studied in the literature. It is not known whether even-hole-free graphs can be optimally colored in polynomial time. A diamond is the graph obtained from the clique of four vertices by removing one edge. A kite is a graph consists of a diamond and another vertex adjacent to a vertex of degree two in the diamond. Kloks et al. (J Combin Theory B 99:733-800, 2009) designed a polynomial-time algorithm to color an (even hole, diamond)-free graph. In this paper, we consider the class of (even hole, kite)-free graphs which generalizes (even hole, diamond)-free graphs. We prove that a connected (even hole, kite)-free graph is diamond-free, or the join of a clique and a diamond-free graph, or contains a clique cutset. This result, together with the result of Kloks et al., imply the existence of a polynomial time algorithm to color (even hole, kite)-free graphs. We also prove (even hole, kite)-free graphs are β-perfect in the sense of Markossian, Gasparian and Reed. Finally, we establish the Vizing bound for (even hole, kite)-free graphs.

International Journal of Pure and Apllied Mathematics, 2015

In this paper, we define the edge C k signed graph of a given signed graph and shown that for any signed graph Σ, its edge C k signed graph E k (Σ) is balanced. We then give structural characterization of edge C k signed graphs. Further, we obtain a structural characterization of signed graphs that are switching equivalent to their the edge C k signed graphs.

Journal of Graph Theory, 2017

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our O(nm)-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our O(n 2.5 + nm)-time algorithm to optimally color them. Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 times the clique number.

Bulletin of the Malaysian Mathematical Sciences Society, 2015

A signed graph is a graph in which every edge is designated to be either positive or negative; it is balanced if every cycle contains an even number of negative edges. A marked signed graph is a signed graph each vertex of which is designated to be positive or negative, and it is consistent if every cycle in the signed graph possesses an even number of negative vertices. Signed line graph L(S) of a given signed graph S = (G, σ), as given by Behzad and Chartrand [9], is the signed graph with the standard line graph L(G) of G as its underlying graph and whose edges are assigned the signs according to the rule: for any e i e j ∈ E(L(S)), e i e j ∈ E − L(S)⇔ the edges e i and e j of S are both negative in S. Iterated signed line graphs L k (S)=L(L k−1 (S)), k ∈ N , S:= L 0 (S) is defined similarly. Further, L(S) is S−consistent if to each vertex e of L(S), which is an edge of S, one assigns the sign σ(e) then the resulting marked signed graph (L(S)) µ is consistent. In this paper, we give a characterization of signed graphs S whose iterated signed line graphs L k (S) are balanced or S−consistent.

2021

A signed graph is an ordered pair Σ = (G, σ), where G = (V,E) is the underlying graph of Σ with a signature function σ : E → {1,−1}. In this article, we define the nth power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an nth power of a signed graph to be unique. Also, we characterize balanced power signed graphs.