Flow number and circular flow number of signed cubic graphs

Let $\mathrm{\Phi }(G,\sigma )$ and ${\mathrm{\Phi }}_c(G,\sigma )$ denote the flow number and the circular flow number of a flow-admissible signed graph $(G,\sigma )$, respectively. It is known that $\mathrm{\Phi }(G)=?{\mathrm{\Phi }}_c(G)?$ for every unsigned graph G. Based on this fact, in 2011 Raspaud and Zhu conjectured that $\mathrm{\Phi }(G,\sigma )?{\mathrm{\Phi }}_c(G,\sigma )<1$ holds also for every flow-admissible signed graph $(G,\sigma )$. This conjecture was disproved by Schubert and Steffen using graphs with bridges and vertices of large degree. In this paper we focus on cubic graphs, since they play a crucial role in many open problems in graph theory. For cubic graphs we show that $\mathrm{\Phi }(G,\sigma )=3$ if and only if ${\mathrm{\Phi }}_c(G,\sigma )=3$ and if $\mathrm{\Phi }(G,\sigma )\in \{4,5\}$, then $4\le {\mathrm{\Phi }}_c(G,\sigma )\le \mathrm{\Phi }(G,\sigma )$. We also prove that all pairs of flow number and circular flow number that fulfil these conditions can be achieved in the family of bridgeless cubic graphs and thereby disprove the conjecture of Raspaud and Zhu even for bridgeless signed cubic graphs. Finally, we prove that all currently known flow-admissible graphs without nowhere-zero 5-flow have flow number and circular flow number 6 and propose several conjectures in this area.     

The signed domatic number of some regular graphs

Let G be a finite and simple graph with vertex set V(G), and let f:V(G)→{−1,1} be a two-valued function. If ∑_x∈N[v]f(x)≥1 for each v∈V(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f₁,f₂,…,f_d} of signed dominating functions on G with the property that for each x∈V(G), is called a signed dominating family (of functions) on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus C_p×C_q.     

Acyclic chromatic indices of planar graphs with large girth

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^′(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.In this paper, we prove that every planar graph G with girth g(G) and maximum degree Δ has a^′(G)=Δ if there exists a pair (k,m)∈{(3,11),(4,8),(5,7),(8,6)} such that G satisfies Δ≥k and g(G)≥m.     

Circular flow on signed graphs

The circular flow number Φc(G,σ) of a signed graph (G,σ) is the minimum r for which an orientation of (G,σ) admits a circular r-flow. We prove that the circular flow number of a signed graph (G,σ) is equal to the minimum imbalance ratio of an orientation of (G,σ). We then use this result to prove that if G is 4-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) (as well as Φ(G,σ)) is at most 4. If G is 6-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) is strictly less than 4.     

Distance graphs with maximum chromatic number

The distance graph G(D) has the set of integers as vertices and two vertices are adjacent in G(D) if their difference is contained in the set D⊂Z. A conjecture of Zhu states that if the chromatic number of G(D) achieves its maximum value |D|+1 then the graph has a triangle. The conjecture is proven to be true if |D|?3. We prove that the chromatic number of a distance graph with D={a,b,c,d} is five only if either D={1,2,3,4k} or D={a,b,a+b,b-a}. This confirms a stronger version of Zhu's conjecture for |D|=4, namely, if the chromatic number achieves its maximum value then the graph contains K₄.     

Game chromatic number of outerplanar graphs

This note proves that the game chromatic number of an outerplanar graph is at most 7. This improves the previous known upper bound of the game chromatic number of outerplanar graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 67–70, 1999     

On the chromatic number of random graphs

In this paper we consider the classical Erdős–Rényi model of random graphs G_n,p. We show that for p=p(n)n^−3/4−δ, for any fixed δ>0, the chromatic number χ(G_n,p) is a.a.s. ℓ, ℓ+1, or ℓ+2, where ℓ is the maximum integer satisfying 2(ℓ−1)log(ℓ−1)p(n−1).     

On the chromatic number of circulant graphs

Given a set D of a cyclic group C, we study the chromatic number of the circulant graph G(C,D) whose vertex set is C, and there is an edge ij whenever i−j∈D∪−D. For a fixed set D={a,b,c:a<b<c} of positive integers, we compute the chromatic number of circulant graphs G(Z_N,D) for all N≥4bc. We also show that, if there is a total order of D such that the greatest common divisors of the initial segments form a decreasing sequence, then the chromatic number of G(Z,D) is at most 4. In particular, the chromatic number of a circulant graph on Z_N with respect to a minimum generating set D is at most 4. The results are based on the study of the so-called regular chromatic number, an easier parameter to compute. The paper also surveys known results on the chromatic number of circulant graphs.     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

The chromatic number and rank of the complements of the Kasami graphs

We determine the rank and chromatic number of the complements of all Kasami graphs, some of which form an infinite family of counterexamples to the now disproven rank-coloring conjecture.     

Strong edge-coloring for planar graphs with large girth

A strong edge-coloring of a graph $G=(V,E)$ is a partition of its edge set $E$ into induced matchings. Let $G$ be a connected planar graph with girth $k\ge 26$ and maximum degree $\Delta$. We show that either $G$ is isomorphic to a subgraph of a very special $\Delta$-regular graph with girth $k$, or $G$ has a strong edge-coloring using at most $2\Delta +\lceil \frac{12(\Delta -2)}{k}\rceil$ colors.     

Six-flows on almost balanced signed graphs

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero 6-flow. By Seymour's 6-flow theorem, Bouchet's conjecture holds for signed graphs with all edges positive. Recently, Rollová et al proved that every flow-admissible signed cubic graph with two negative edges admits a nowhere-zero 7-flow, and admits a nowhere-zero 6-flow if its underlying graph either contains a bridge, or is 3-edge-colorable, or is critical. In this paper, we improve and extend these results, and confirm Bouchet's conjecture for signed graphs with frustration number at most two, where the frustration number of a signed graph is the smallest number of vertices whose deletion leaves a balanced signed graph.     

Lower bounds on the signed domination numbers of directed graphs

A numerical invariant of directed graphs concerning domination which is named signed domination number γ_S is studied in this paper. We present some sharp lower bounds for γ_S in terms of the order, the maximum degree and the chromatic number of a directed graph.