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This paper proves that every planar graph without cycles of length 4, 7, or 9 is DP-3-colorable., Comment: 12 pages, 2 figures

A graph $G$ is $(I,F)$-partitionable if its vertex set can be partitioned into two parts such that one part is an independent set, and the other induces a forest. In this paper, we prove that every planar graph without cycles of length $4, 6, 9$ is $(I,F)$-partitionable., Comment: 9 pages, 5 figures

The study on parity signed graphs was initiated by Acharya and Kureethara very recently and then followed by Zaslavsky etc.. Let $(G,\sigma)$ be a signed graph on $n$ vertices. If $(G,\sigma)$ is switch-equivalent to $(G,+)$ at a set of $\lfloor \frac{n}{2} \rfloor$ many vertices, then we call $(G,\sigma)$ a parity signed graph and $\sigma$ a parity-signature. $\Sigma^{-}(G)$ is defined as the set of the number of negative edges of $(G,\sigma)$ over all possible parity-signatures $\sigma$. The $rna$ number $\sigma^-(G)$ of $G$ is given by $\sigma^-(G)=\min \Sigma^{-}(G)$. In other words, $\sigma^-(G)$ is the smallest cut size that has nearly equal sides. In this paper, all graphs considered are finite, simple and connected. We apply switch method to the characterization of parity signed graphs and the study on the $rna$ number. We prove that: for any graph $G$, $\Sigma^{-}(G)=\left\{\sigma^{-}(G)\right\}$ if and only if $G$ is $K_{1, n-1} $ with $n$ even or $K_{n}$. This confirms a conjecture proposed in [M. Acharya and J.V. Kureethara. Parity labeling in signed graphs. J. Prime Res. Math., to appear. arXiv:2012.07737]. Moreover, we prove a nontrivial upper bound for the $rna$ number: for any graph $G$ on $m$ edges and $n$ ($n\geq 4$) vertices, $\sigma^{-}(G)\leq \lfloor \frac{m}{2}+\frac{n}{4} \rfloor$. We show that $K_n$, $K_n-e$ and $K_n-\triangle$ are the only three graphs reaching this bound. This is the first upper bound for the $rna$ number so far. Finally, we prove that: for any graph $G$, $\sigma^-(G)+\sigma^-(\overline{G})\leq \sigma^-(G\cup \overline{G})$, where $\overline{G}$ is the complement of $G$. This solves a problem proposed in [M. Acharya, J.V. Kureethara and T. Zaslavsky. Characterizations of some parity signed graphs. 2020, arXiv:2006.03584v3]., Comment: 10 pages

A graph $G$ is $(d_1,d_2,d_3)$-colorable if the vertex set $V(G)$ can be partitioned into three subsets $V_1,V_2$ and $V_3$ such that for $i\in\{1,2,3\}$, the induced graph $G[V_i]$ has maximum vertex-degree at most $d_i$. So, $(0,0,0)$-colorability is exactly 3-colorability. The well-known Steinberg's conjecture states that every planar graph without cycles of length 4 or 5 is 3-colorable. As this conjecture being disproved by Cohen-Addad etc. in 2017, a similar question, whether every planar graph without cycles of length 4 or $i$ is 3-colorable for a given $i\in \{6,\ldots,9\}$, is gaining more and more interest. In this paper, we consider this question for the case $i=6$ from the viewpoint of improper colorings. More precisely, we prove that every planar graph without cycles of length 4 or 6 is (1,0,0)-colorable, which improves on earlier results that they are (2,0,0)-colorable and also (1,1,0)-colorable, and on the result that planar graphs without cycles of length from 4 to 6 are (1,0,0)-colorable., Comment: 21 pages, 1 figure

In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter $\mu_3$ is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $|E(G)|-\mu_3(G)$, which is no less than $\frac{4}{5}|E(G)|$, many edges are normal. This result improves on some earlier results of B\'{\i}lkov\'{a} and \v{S}\'{a}mal., Comment: 14 pages, 8 pages, an earlier version is included in the dissertation of the first author by Universitaet Paderborn

The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(K_q,+)$ by repeatedly applying these operations. This result gives a signed version of the Haj\'os theorem, emphasizing the role of all-positive complete graphs played in the class of signed graphs, as played in the class of unsigned graphs., Comment: 11 pages, 2 figures

Listed as No. 53 among the one hundred famous unsolved problems in [J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, Berlin, 2008] is Steinberg's conjecture, which states that every planar graph without 4- and 5-cycles is 3-colorable. In this paper, we show that plane graphs without 4- and 5-cycles are 3-colorable if they have no ext-triangular 7-cycles. This implies that (1) planar graphs without 4-, 5-, 7-cycles are 3-colorable, and (2) planar graphs without 4-, 5-, 8-cycles are 3-colorable, which cover a number of known results in the literature motivated by Steinberg's conjecture., Comment: 15 pages, 2 figure. arXiv admin note: text overlap with arXiv:1506.04629

The only open case of Vizing's conjecture that every planar graph with $\Delta\geq 6$ is a class 1 graph is $\Delta = 6$. We give a short proof of the following statement: there is no 6-critical plane graph $G$, such that every vertex of $G$ is incident to at most three 3-faces. A stronger statement without restriction to critical graphs is stated in \cite{Wang_Xu_2013}. However, the proof given there works only for critical graphs. Furthermore, we show that every 5-critical plane graph has a 3-face which is adjacent to a $k$-face $(k\in \{3,4\})$. For $\Delta = 5$ our result gives insights into the structure of planar $5$-critical graphs, and the result for $\Delta=6$ gives support for the truth of Vizing's planar graph conjecture., Comment: 4 pages, 1 figure

The chromatic number $\chi((G,\sigma))$ of a signed graph $(G,\sigma)$ is the smallest number $k$ for which there is a function $c : V(G) \rightarrow \mathbb{Z}_k$ such that $c(v) \not= \sigma(e) c(w)$ for every edge $e = vw$. Let $\Sigma(G)$ be the set of all signatures of $G$. We study the chromatic spectrum $\Sigma_{\chi}(G) = \{\chi((G,\sigma))\colon\ \sigma \in \Sigma(G)\}$ of $(G,\sigma)$. Let $M_{\chi}(G) = \max\{\chi((G,\sigma))\colon\ \sigma \in \Sigma(G)\}$, and $m_{\chi}(G) = \min\{\chi((G,\sigma))\colon\ \sigma \in \Sigma(G)\}$. We show that $\Sigma_{\chi}(G) = \{k : m_{\chi}(G) \leq k \leq M_{\chi}(G)\}$. We also prove some basic facts for critical graphs. Analogous results are obtained for a notion of vertex-coloring of signed graphs which was introduced by M\'{a}\v{c}ajov\'{a}, Raspaud, and \v{S}koviera., Comment: 6 pages

Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $\chi_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number $\chi$. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are signed graphs where the difference is 1. On the other hand, for a signed graph on $n$ vertices, if the difference is smaller than 1, then there exists $\epsilon_n>0$, such that the difference is at most $1 - \epsilon_n$. We also show that notion of $(k,d)$-colorings is equivalent to $r$-colorings (see (X. Zhu, Recent developments in circular coloring of graphs, in Topics in Discrete Mathematics Algorithms and Combinatorics Volume 26, Springer Berlin Heidelberg (2006) 497-550))., Comment: 16 pages, 2 figures

In this paper, we prove that planar graphs without cycles of length 4, 6, 9 are 3-colorable., Comment: 15 pages, 1 figure

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each $k \in \{3,4,5,6\}$, every signed planar graph without circuits of length $k$ is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of length 4 is 3-choosable. We construct a signed planar graph with girth 4 which is not 3-choosable but the unsigned graph is 3-choosable., Comment: We updated the reference list

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs, based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist. For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs., Comment: This is new version with updated title and abstract. Furthermore, in Lemma 10, the proof of the families of graphs $G_n, H_n, T_n$ to be critical is added. The published version of the article can be found in "The electronic journal of combinatorics"