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Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A bijection $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w(u)\ne w(v)$, where $w(u) = f(u) + \sum_{e\in E(u)} f(e)$ and $E(u)$ is the set of incident edge(s) of $u$. The local antimagic total chromatic number, denoted $\chi_{lat}(G)$, is the minimum number of distinct weights over local antimagic total labeling of $G$. In this paper, we provide a correct proof and exact local antimagic total chromatic number of path and spider graphs given in [Local vertex antimagic total coloring of path graph and amalgamation of path, {\it CGANT J. Maths Appln.} {\bf 5(1)} 2024, DOI:10.25037/cgantjma.v5i1.109]. Further, we determined the local antimagic total chromatic number of spider graph with each leg of length at most 2. We also showed the existence of unicyclic and bicyclic graphs with local antimagic total chromatic number 3.

The locating rainbow connection number of a graph is defined as the minimum number of colors required to color vertices such that every two vertices there exists a rainbow vertex path and every vertex has a distinct rainbow code. This rainbow code signifies a distance between vertices within a given set of colors in a graph. This paper aims to determine the locating rainbow connection number for vertex-transitive graphs. Three main theorems are derived, focusing on the locating rainbow connection number for some vertex-transitive graphs, Comment: The article omprises a total of 11 pages and includes a total of 9 figures

A radio labeling of a graph $G$ is a function $f : V(G) \rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest number $k$ such that $G$ has radio labeling $f$ with max$\{f(v):v \in V(G)\} = k$. In this paper, we give a lower bound for the radio number for the Cartesian product of the generalized Petersen graph and tree. We present two necessary and sufficient conditions, and three other sufficient conditions to achieve the lower bound. Using these results, we determine the radio number for the Cartesian product of the Peterson graph and stars., Comment: 15 pages, 1 figure

In this paper, we provide a correct proof for the lower bounds of the local antimagic chromatic number of the corona product of friendship and fan graphs with null graph respectively as in [On local antimagic vertex coloring of corona products related to friendship and fan graph, {\it Indon. J. Combin.}, 5(2) (2021) 110--121]. Consequently, we obtained a sharp lower bound that gives the exact local antimagic chromatic number of the corona product of friendship and null graph.

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $g^+(u) \ne g^+(v)$, where $g^+(u) = \sum_{e\in E(u)} g(e)$, and $E(u)$ is the set of edges incident to $u$. Similarly, a bijection $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w_f(u)\ne w_f(v)$, where $w_f(u) = f(u) + \sum_{e\in E(u)} f(e)$. Thus, any local antimagic (total) labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $g^+(v)$ (respectively, $w_f(u)$). The local antimagic (total) chromatic number, denoted $\chi_{la}(G)$ (respectively $\chi_{lat}(G)$), is the minimum number of induced colors taken over local antimagic (total) labeling of $G$. In this paper, we determined $\chi_{lat}(G)$ where $G$ is the amalgamation of complete graphs.

A radio labelling of a graph $G$ is a mapping $f : V(G) \rightarrow \{0, 1, 2,\ldots\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number $rn(G)$ of $G$ is the smallest integer $k$ such that $G$ admits a radio labelling $f$ with $\max\{f(v):v \in V(G)\} = k$. The weight of a tree $T$ from a vertex $v \in V(T)$ is the sum of the distances in $T$ from $v$ to all other vertices, and a vertex of $T$ achieving the minimum weight is called a weight center of $T$. It is known that any tree has one or two weight centers. A tree is called a two-branch tree if the removal of all its weight centers results in a forest with exactly two components. In this paper we obtain a sharp lower bound for the radio number of two-branch trees which improves a known lower bound for general trees. We also give a necessary and sufficient condition for this improved lower bound to be achieved. Using these results, we determine the radio number of two families of level-wise regular two-branch trees., Comment: 29 pages, 3 figures

Let $G$ be a simple finite connected graph of order $n$. The detour distance between two distinct vertices $u$ and $v$ denoted by $D(u,v)$ is the length of a longest $uv$-path in $G$. A hamiltonian coloring $h$ of a graph $G$ of order $n$ is a mapping $h : V(G) \rightarrow \{0,1,2,...\}$ such that $D(u,v) + |h(u)-h(v)| \geq n-1$, for every two distinct vertices $u$ and $v$ of $G$. The span of $h$, denoted by $span(h)$, is $\max\{|h(u)-h(v)| : u, v \in V(G)\}$. The hamiltonian chromatic number of $G$ is defined as $hc(G) := \min\{span(h)\}$ with minimum taken over all hamiltonian coloring $h$ of $G$. In this paper, we give an improved lower bound for the hamiltonian chromatic number of trees and give a necessary and sufficient condition to achieve the improved lower bound. Using this result, we determine the hamiltonian chromatic number of two families of trees., Comment: This is a final version appeared in proceedings of RAGT 2019 Conference

A graph $G = (V, E)$ of order $p$ and size $q$ is said to be local antimagic if there exists a bijection $g:E(G) \to \{1,2,\ldots,q\}$ such that for any pair of adjacent vertices $u$ and $v$, $g^+(u)\ne g^+(v)$, where $g^+(u)=\sum_{uv\in E(G)} g(uv)$ is the induced vertex color of $u$ under $g$. We also say $G$ is local antimagic total if there exists a bijection $f: V\cup E \to\{1,2,\ldots ,p+q\}$ such that for any pair of adjacent vertices $u$ and $v$, $w(u)\not= w(v)$, where $w(u)= f(u) +\sum_{uv\in E(G)} f(uv)$ is the induced vertex weight of $u$ under $f$. The local antimagic (and local antimagic total) chromatic number of $G$, denoted $\chi_{la}(G)$ (and $\chi_{lat}(G)$), is the minimum number of distinct induced vertex colors (and weights) over all local antimagic (and local antimagic total) labelings of $G$. We also say a local antimagic total labeling is local super antimagic total if $f(v)\in\{1,2,\ldots,p\}$ for each $v\in V(G)$. In [Proof of a local antimagic conjecture, {\it Discrete Math. Theor. Comp. Sc.}, {\bf 20(1)} (2018), \#18], the author proved that every connected graph of order at least 3 is local antimagic. Using this result, we provide a very short proof that every graph is local antimagic total. We showed that there exists close relationship between $\chi_{la}(G \vee K_1)$ and $\chi_{lat}(G)$. A sufficient condition is also given for the corresponding local super antimagic total labeling. Sharp bounds of $\chi_{lat}(G)$ and close relationships between $\chi_{lat}(G)$ and $\chi_{la}(G \vee K_1)$ are found. Bounds of $\chi_{lat}(G-e)$ in terms of $\chi_{lat}(G)$ for a graph $G$ with an edge $e$ deleted are also obtained. These relationships are used to determine the exact values of $\chi_{lat}$ for many graphs $G$. We also conjecture that each graph $G$ of order at least 3 has $\chi_{lat}(G)\le \chi_{la}(G)$., Comment: It is extended to 27 pages with many more new results

Let $G$ be a simple connected graph of order $n$. A hamiltonian coloring $c$ of a graph $G$ is an assignment of colors (non-negative integers) to the vertices of $G$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n - 1$ for every two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$ denotes the detour distance between $u$ and $v$ in $G$ which is the length of the longest path between $u$ and $v$. The value \emph{hc(c)} of a hamiltonian coloring $c$ is the maximum color assigned to a vertex of $G$. The hamiltonian chromatic number, denoted by $hc(G)$, is min\{$hc(c)$\} taken over all hamiltonian coloring $c$ of $G$. In this paper, we give a necessary and sufficient condition to achieve a lower bound for the hamiltonian chromatic number of block graphs given in [Theorem 1,On Hamiltonian Colorings of Block graphs, In: Kaykobad, M., Petrechi, R., (eds.) WALCOM: Algorithms and Computation, LNCS: 9627, 28-39, 2016]. We present an algorithm for optimal hamiltonian coloring of a special class of block graphs, namely $SDB(p/2)$ block graphs. We characterize level-wise regular block graphs and extended star of blocks achieving this lower bound., Comment: This is the final version appeared in Journal of Graph Algorithms and Applications

A Stacked-book graph $G_{m,n}$ results from the Cartesian product of a star graph $S_m$ and path $P_n$, where $m$ and $n$ are the orders of $S_m$ and $P_n$ respectively. A radio labeling problem of a simple and connected graph, $G$, involves a non-negative integer function $f:V(G)\rightarrow \mathbb Z^+$ on the vertex set $V(G)$ of G, such that for all $u,v \in V(G)$, $|f(u)-f(v)| \geq \textmd{diam}(G)+1-d(u,v)$, where $\textmd {diam}(G)$ is the diameter of $G$ and $d(u,v)$ is the shortest distance between $u$ and $v$. Suppose that $f_{min}$ and $f_{max}$ are the respective least and largest values of $f$ on $V(G)$, then, span$f$, the absolute difference of $f_{min}$ and $f_{max}$, is the span of $f$ while the radio number $rn(G)$ of $G$ is the least value of span$f$ over all the possible radio labels on $V(G)$. In this paper, we obtain the radio number for the stacked-book graph $G_{m,n}$ where $m \geq 4$ and $n$ is even, and obtain bounds for $m=3$ which improves existing upper and lower bounds for $G_{m,n}$ where $m=3$., Comment: 9 pages, 2 figures

A proper total $k$-colouring of a graph $G=(V,E)$ is an assignment $c : V \cup E\to \{1,2,\ldots,k\}$ of colours to the edges and the vertices of $G$ such that no two adjacent edges or vertices and no edge and its end-vertices are associated with the same colour. A total neighbour sum distinguishing $k$-colouring, or tnsd $k$-colouring for short, is a proper total $k$-colouring such that $\sum_{e\ni u}c(e)+c(u)\neq \sum_{e\ni v}c(e)+c(v)$ for every edge $uv$ of $G$. We denote by $\chi''_{\Sigma}(G)$ the total neighbour sum distinguishing index of $G$, which is the least integer $k$ such that a tnsd edge $k$-colouring of $G$ exists. It has been conjectured that $\chi''_{\Sigma}(G) \leq \Delta(G) + 3$ for every graph $G$. In this paper we confirm this conjecture for any graph $G$ with ${\rm mad}(G)<\frac{14}{3}$ and $\Delta(G) \geq 8$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1508.06112

An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \& Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \& Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than $K_2$ ., Comment: Final version for publication in DMTCS. Changes from previous version are formatting to journal style and correction of two minor typographical errors

A hamiltonian coloring $c$ of a graph $G$ of order $n$ is a mapping $c$ : $V(G) \rightarrow \{0,1,2,...\}$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n-1$, for every two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$ denotes the detour distance between $u$ and $v$ which is the length of a longest $u,v$-path in $G$. The value $hc(c)$ of a hamiltonian coloring $c$ is the maximum color assigned to a vertex of $G$. The hamiltonian chromatic number, denoted by $hc(G)$, is the min{$hc(c)$} taken over all hamiltonian coloring $c$ of $G$. In this paper, we present a lower bound for the hamiltonian chromatic number of trees and give a sufficient condition to achieve this lower bound. Using this condition we determine the hamiltonian chromatic number of symmetric trees, firecracker trees and a special class of caterpillars., Comment: 12 pages, CALDAM 2016 conference proceeding paper

A radio labeling of a graph $G$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that $|f(u)-f(v)|\geq d + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$, where $d$ is the diameter of $G$ and $d(u,v)$ the distance between $u$ and $v$ in $G$. The radio number of $G$ is the smallest integer $k$ such that $G$ has a radio labeling $f$ with $\max\{f(v) : v \in V(G)\} = k$. We give a necessary and sufficient condition for a lower bound on the radio number of trees to be achieved, two other sufficient conditions for the same bound to be achieved by a tree, and an upper bound on the radio number of trees. Using these, we determine the radio number for three families of trees., Comment: 19 pages, 7 figures. This is the final version accepted in Discrete Applied Mathematics

A hamiltonian coloring c of a graph G of order p is an assignment of colors to the vertices of G such that $D(u,v)+|c(u)-c(v)|\geq p-1$ for every two distinct vertices u and v of G, where D(u,v) denoted the detour distance between u and v. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we present a lower bound for the hamiltonian chromatic number of block graphs and give a sufficient condition to achieve the lower bound. We characterize symmetric block graphs achieving this lower bound. We present two algorithms for optimal hamiltonian coloring of symmetric block graphs., Comment: 12 pages, 1 figure. A conference version appeared in the proceedings of WALCOM 2016

A \emph{locally irregular graph} is a graph whose adjacent vertices have distinct degrees. We say that a graph $G$ can be decomposed into $k$ locally irregular subgraphs if its edge set may be partitioned into $k$ subsets each of which induces a locally irregular subgraph in $G$. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree $3$, every connected graph can be decomposed into $3$ locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of sufficiently large minimum degree, $\delta(G)\geq 10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244]., Comment: 12 pages

An $L(2, 1)$-labeling of a graph $G$ is an assignment of a nonnegative integer to each vertex of $G$ such that adjacent vertices receive integers that differ by at least two and vertices at distance two receive distinct integers. The span of such a labeling is the difference between the largest and smallest integers used. The $\lambda$-number of $G$, denoted by $\lambda(G)$, is the minimum span over all $L(2, 1)$-labelings of $G$. Bodlaender {\it et al.} conjectured that if $G$ is an outerplanar graph of maximum degree $\Delta$, then $\lambda(G)\leq \Delta+2$. Calamoneri and Petreschi proved that this conjecture is true when $\Delta \geq 8$ but false when $\Delta=3$. Meanwhile, they proved that $\lambda(G)\leq \Delta+5$ for any outerplanar graph $G$ with $\Delta=3$ and asked whether or not this bound is sharp. In this paper we answer this question by proving that $\lambda(G)\leq \Delta + 3$ for every outerplanar graph with maximum degree $\Delta=3$. We also show that this bound $\Delta + 3$ can be achieved by infinitely many outerplanar graphs with $\Delta=3$.

Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than $k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of investigating irregularities in graphs, it has been moreover conjectured that only slightly larger $k$, i.e., $k=\Delta(G)+2$ enables enforcing additional strong feature of $c$, namely that it attributes distinct sums of incident colours to adjacent vertices in $G$ if only this graph has no isolated edges and is not isomorphic to $C_5$. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact even stronger statement holds, as the necessary number of colours stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph $G$ of maximum degree at least $28$ which contains no isolated edges admits a proper edge colouring $c:E\to\{1,2,\ldots,\Delta(G)+1\}$ such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$., Comment: 22 pages

Given a finite or infinite graph $G$ and positive integers $\ell, h_1, h_2, h_3$, an $L(h_1, h_2, h_3)$-labelling of $G$ with span $\ell$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots, \ell\}$ such that, for $i = 1, 2, 3$ and any $u, v \in V(G)$ at distance $i$ in $G$, $|f(u) - f(v)| \geq h_i$. A $C(h_1, h_2, h_3)$-labelling of $G$ with span $\ell$ is defined similarly by requiring $|f(u) - f(v)|_{\ell} \ge h_i$ instead, where $|x|_{\ell} = \min\{|x|, \ell-|x|\}$. The minimum span of an $L(h_1, h_2, h_3)$-labelling, or a $C(h_1, h_2, h_3)$-labelling, of $G$ is denoted by $\lambda_{h_1,h_2,h_3}(G)$, or $\sigma_{h_1,h_2,h_3}(G)$, respectively. Two related invariants, $\lambda^*_{h_1,h_2,h_3}(G)$ and $\sigma^*_{h_1,h_2,h_3}(G)$, are defined similarly by requiring further that for every vertex $u$ there exists an interval $I_u$ $\mod~(\ell + 1)$ or $\mod~\ell$, respectively, such that the neighbours of $u$ are assigned labels from $I_u$ and $I_v \cap I_w = \emptyset$ for every edge $vw$ of $G$. A recent result asserts that the $L(2,1,1)$-labelling problem is NP-complete even for the class of trees. In this paper we study the $L(h, p, p)$ and $C(h, p, p)$ labelling problems for finite or infinite trees $T$ with finite maximum degree, where $h \ge p \ge 1$ are integers. We give sharp bounds on $\lambda_{h,p,p}(T)$, $\lambda^*_{h,p,p}(T)$, $\sigma_{h, 1, 1}(T)$ and $\sigma^*_{h, 1, 1}(T)$, together with linear time approximation algorithms for the $L(h,p,p)$-labelling and the $C(h, 1, 1)$-labelling problems for finite trees. We obtain the precise values of these four invariants for a few families of trees. We give sharp bounds on $\sigma_{h,p,p}(T)$ and $\sigma^*_{h,p,p}(T)$ for trees with maximum degree $\Delta \le h/p$, and as a special case we obtain that $\sigma_{h,1,1}(T) = \sigma^*_{h,1,1}(T) = 2h + \Delta - 1$ for any tree $T$ with $\Delta \le h$.